Unit 2-2: Writing and Graphing Quadratics NOTE PACKET. 12. I can use the discriminant to determine the number and type of solutions/zeros.

Transcription

1 Unit 2-2: Writing and Graphing Quadratics NOTE PACKET Name: Period Learning Targets: Unit I can use the discriminant to determine the number and type of solutions/zeros. 1. I can identify a function as quadratic given a table, equation, or graph. Modeling with Quadratic Functions 2. I can determine the appropriate domain and range of a quadratic equation or event. 3. I can identify the minimum or maximum and zeros of a function with a calculator. 4. I can apply quadratic functions to model real-life situations, including quadratic regression models from data. 5. I can graph quadratic functions in standard form (using properties of quadratics). Graphing 6. I can graph quadratic functions in vertex form (using basic transformations). 7. I can identify key characteristics of quadratic functions including axis of symmetry, vertex, min/max, y-intercept, x-intercepts, domain and range. 8. I can rewrite quadratic equations from standard to vertex and vice versa. Writing Equations of Quadratic Functions 9. I can write quadratic equations given a graph or given a vertex and a point (without a calculator). 10. I can write quadratic expressions/functions/equations given the roots/zeros/x-intercepts/solutions. 11. I can write quadratic equations in vertex form by completing the square. Revisit Applications 4R. I can apply quadratics functions to real life situations without using the graphing calculator. 1

2 Unit 2-2 Writing and Graphing Quadratics Worksheets Completed Date LTs Pages Problems Done Quiz/Unit Test Dates(s) Date LTs Score Corrected Retake Quiz Retakes Dates and Rooms 2

3 Modeling Quadratic Functions Date: After this lesson and practice, I will be able to identify a function as quadratic given a table, equation, or graph. (LT 1) determine the appropriate domain and range of a quadratic equation or event. (LT 2) identify the minimum or maximum, and zeros of a function with a calculator. (LT 3) apply quadratic functions to model real-life situations, including quadratic regression models. (LT 4) Let s review the definition of a quadratic function Quadratic Function: A function that can be written in the form where. The domain of a quadratic function is. Standard Form of a Quadratic Function: Example 1: Determine whether each function is quadratic or not. If so, write it in standard form. a. y = x(x- 5) b. c. y= x 2 (5x x 2 ) d. f(x) = (2x+3)(x- 7) Properties of Quadratic Functions Let s graph the parent function for a quadratic : y = x! x y The shape of the graph is called a. The point at which the y value of the graph comes to a maximum or minimum is called the. It is the ordered pair. The vertical line line that passes through the vertex of the parabola is called the. It has the equation. 3

4 Example 2: Suppose a quadratic function as a vertex of (- 3,- 5) and contains the points (- 4,- 3) and (- 2,3). Identify the equation for the axis of symmetry and use it to help you sketch a graph of the function. Then identify the domain and range. Example 3: Suppose a quadratic function as a vertex of (5,6) and contains the points (7,2) and (2,- 3). Identify the equation for the axis of symmetry and use it to help you sketch a graph of the function. Then identify the domain and range. The places where the graph crosses the x - axis are called: Using Your Calculator (LT 3) In some situations, you want to quickly approximate the and the / of a quadratic equation. Thankfully, your graphing calculator is very helpful in accomplishing this feat. Example 4: Approximate the zeros and min or max of the equation 3x 2 + 5x = 6+y using a graphing calculator. Round to four decimal places. Approximate the zeros Note: This method will not work if the zero is a min or max. 1) Put the equation you want to graph in standard form. 2) Enter the equation in your calculator as Y1 =. Press GRAPH. (Adjust your window if needed. ZOOM ZStandard gives 10 by 10 window) 3) Press 2 nd TRACE, then press 2: ZERO. 4) Left Bound? Move your cursor just to the left of the first point of intersection. Press ENTER. 5) Right Bound? Move your cursor just to the right of the first point of intersection. Press ENTER. 6) The screen will show Guess. Press ENTER again. 7) The bottom of the screen will say X= Y=0. The x value is the zero. The first zero in this example is. 8) Repeat steps 3-7 to obtain the second root. The second zero is. 4

5 If you are having trouble using the ZERO function try this to approximate zeros: 1) Put the equation you want to graph in standard form. 2) Put equation into Y1 and y=0 into Y2. 3) Press 2 nd TRACE, then press 5: INTERSECT 4) First curve? Move your cursor close to the point of intersection. Press ENTER. 5) Second Curve? If needed move your cursor close to the point of intersection. Press ENTER. 6) The screen will show Guess. Press ENTER again. 7) The bottom of the screen will say X= Y=0. The x value is the zero. The first zero in this example is. 8) Repeat steps 3-7 to obtain the second root. The second zero is. Approximate the min or max 1) Press 2 nd TRACE, then press MIN or MAX (depending on the shape of your parabola). 2) Move your cursor just to the left of the min or max. Press ENTER. 3) Move your cursor just to the right of the min or max. Press ENTER. 4) The screen will show Guess. Press ENTER again. 5) The bottom of the screen will say X= Y = The y value is the min or max. The x value is where the min or max is occurring. The min or max in this example is at. Example 5: Approximate the zeros and min or max of the equation below using a graphing calculator. Round to four decimal places. a) 2x 2 +10x 1= y b) f (x)= x 2 +8x +16 Zero(s) : Min Max is at Zero(s) : Min Max is at 5

6 Applications of Quadratic Functions (LT 4) Quadratic functions have many applications to real- world scenarios. For example, quadratics can be used to model the height of a projectile over time. They can also be used to help businesses predict revenue. Example 6: The height of a punted football can be modeled by h = 0.01x x +2. The horizontal distance in feet from the punter s foot is x, and h is the height of the ball in feet. Calculate how far the punter kicked the ball using your graphing calculator. feet yards Example 7: Smoke jumpers are firefighters who parachute into areas near forest fires. A jumper s height during free fall (before the parachute opens) is modeled by h(t ) = 16t 2 + h0 where t is time in seconds and h0 is the initial height. If a jumper s plane is!! 1500 feet off the ground when he jumps, how long is he in free fall before his parachute opens at 1200 feet? Example 8: An apartment complex has 100 two- bedroom units. The monthly profit in dollars realized from renting out the apartments is given by: P(x) = - 10x x 20,000 a. How many units must be rented out to maximize profits? b. What is the maximum profit realizable? c. How many apartments must be rented before a profit is made? Example 9 : A bakery knows from market research that - 6p models the number of pies sold on average day, where p is the price of a pie in dollars. a. Write an equation to represent the revenue the bakery can expect on an average day. R(p) = b. What price will maximize the revenue? c. What is the maximum revenue? 6

7 Example 10 If Jimmy Fallon dropped a T.V. from a height of 1500 feet, how long would it take to hit the ground? How long would it take to land on a building 50 feet tall. Use the formula: h(t) = - 16t 2 + s h(t) = h = height when the object landed. If it landed on the ground, then h =. If it landed on the top of a 50 ft. building, then h =. t = the time it takes for the object to fall. s = height from which the object is dropped or. where the object started! time to ground time to building Example 11: The data in the table represents the amount of rainfall for each month, in inches. a. Graph a scatter plot of the data in your calculator. What type of function best represents the data? - Enter your data (2 ND STAT, Edit ) - Turn on Stat Plot 1 (2 nd, STAT PLOT) - Graph the data (adjust window if needed) b. Find the regression line that models the data using your calculator. - - STAT, calc, QuadReg Look at R 2 value to see if it is a good fit R(x) = Month Rainfall (inches) Jan 5.7 Feb 4.2 Mar 3.8 Apr 2.4 May 1.7 Jun 1.6 Jul 0.8 Aug 1 Sep 1.8 Oct 2.1 Nov??? Dec 5.4 Type equation into Y1 or use directions below: Stat Calc QUADReg VARS Y- VARS Function Y1 Now Graph and see how line matches scatterplot c. Using your equation, predict the rainfall in November. R(11) = 7

8 Graphing with Properties of Quadratics Date: After this lesson and practice, I will be able to graph quadratic functions in standard form (using properties of quadratics). (LT 5) identify key characteristics of quadratic functions including axis of symmetry, vertex, min/max, y- intercept, x-intercepts, domain and range (LT 7) One of the most important things we will do this year is graph functions. The simplest (but not always easiest way) of graphing is to make a table Example 1: Graph the quadratic function by making a table, then write the axis of symmetry, the domain, and the range: y = - 2x Now let s explore some easier ways of graphing. Start with the parent quadratic function: 8

9 Go to and click Start Graphing. In the first blank, type in the parent function (use the carrot key ^ for the squared part of the equation). You should see a red parabola appear with its vertex at (0,0). Follow the directions below and fill in the blanks. In the box where you just wrote the parent function, add an a before the x. You should see a blue button appear next to the words add slider. Click the blue button. You should now be able to change the value of a by using the slider. Experiment with the slider, then fill in the blanks When a > 0, the parabola opens. When a < 0, the parabola opens. Go back to the parent function and add +bx. The first box should now read: y = ax! + bx. Add a slider for b like you did with a, then experiment with moving b around. In Box 4, add the following equation: x =!!!. The fraction will automatically form when you type in /. You should see a different colored line appear on the graph. Now experiment with the a and b sliders. x =!!! is the. (It is an equation x = ) So the x- coordinate of the vertex is! In other words, the vertex is:!!!, f(!!!!! and the y- coordinate is what you get when you plug in! ) (It is an ordered pair.) Finally, add +c to the parent function in Box 1. It should now read: y = ax! + bx + c. Add a slider for c, then experiment with moving a, b, and c. The point (0, c) is the. These properties make graphing much easier! Graph the following functions on your calculator y = 3x 2 5 y = 0.5x 2 + 2x + 3 y = - x 2 + 3x + 7 y = - 0.2x 2 x + 8!!. 9

10 Example 2: Graph y = x 2 2x - 3 (Identify a, b, and c? a= b= c= Steps to graphing quadratic functions: 1. Find the equation of the axis of symmetry and graph. Equation of Axis of Symmetry: 2. Find the order pair of the vertex and graph. Order Pair of Vertex 3. Identify the y- intercept. State and graph the ordered pair. Graph the point and its point of reflection across the axis of symmetry. Symmetry Point 4. Identify the x- intercepts. Set the equations = to zero and find the x- intercepts (or zeros). Use factoring, quadratic formula or completing the square. State and graph the point(s). 5. Evaluate the function for another value of x. Plot the point and its point of reflection across the axis of symmetry. Generally, choose points one less or greater than the x value of the vertex and then choose one two less or greater than the x value of the vertex. Make a table to display values plotted 6. Draw a through the graphed points. 10

11 Examples 3 and 4 3. Graph y = x 2 2x +1 Equation of Axis of Symmetry: x = What is the vertex? (, ) 4. y = x2 + 4x + 5 Equation of Axis of Symmetry: x = What is the vertex? (, ) y- intercept: ( 0, ) Does this function have a min or max (circle one)? What is the minimum or maximum function value? y = Where does the min or max occur? x = What are the domain and range? Domain: Range : x- intercepts and Find 2 points around the vertex and another 2 points using symmetry. y- intercept: ( 0, ) Does this function have a min or max (circle one)? What is the minimum or maximum function value? y = Where does the min or max occur? x = What are the domain and range? Domain: Range : x- intercepts and Find 2 points around the vertex and another 2 points using symmetry. 11

12 Examples 5 and 6 5. Graph y = - x 2 + 4x y = (1/ 2)x 2 + 4x +10 Equation of Axis of Symmetry: x = What is the vertex? (, ) Equation of Axis of Symmetry: x = What is the vertex? (, ) y- intercept: ( 0, ) Does this function have a min or max (circle one)? What is the minimum or maximum function value? y = Where does the min or max occur? x = What are the domain and range? Domain: Range : x- intercepts and Find 2 points around the vertex and another 2 points using symmetry. y- intercept: ( 0, ) Does this function have a min or max (circle one)? What is the minimum or maximum function value? y = Where does the min or max occur? x = What are the domain and range? Domain: Range : x- intercepts and Find 2 points around the vertex and another 2 points using symmetry. 12

13 Ex. 7) y = x 2 2x + 5 a =, b =, c = Opens: because Axis of Symmetry: Vertex: Min Max of at Domain: Range: X- intercepts: Y- intercept: Ex. 8) y = 5x 2 2x + 1 a =, b =, c = Opens: because Axis of Symmetry: Vertex: Min Max of at Domain: Range: X- intercepts: Y- intercept: Ex. 9) y = 1 3 x 2 x a =, b =, c = Opens: because Axis of Symmetry: Vertex: Min Max of at Domain: Range: X- intercepts: Y- intercept: 13

14 More on Graphing Parabolas 1. y = x 2 + 4x 2 Opens: because Axis of Symmetry: Vertex: Domain: Min Max of at Range: X- intercepts: Y- intercept: 2. y = 2x 2 + 4x 1 Opens: because Axis of Symmetry: Vertex: Domain: Min Max of at Range: X- intercepts: Y- intercept: 3. y = x 2 4 Opens: because Axis of Symmetry: Vertex: Domain: Min Max of at Range: X- intercepts: Y- intercept: 4. y = 1 4 x2 x +3 Opens: because Axis of Symmetry: Vertex: Domain: Min Max of at Range: X- intercepts: Y- intercept: 14

15 Graphing with Transformations Date: After this lesson and practice, I will be able to graph quadratic functions in vertex form (using basic transformations). (LT 6) identify key characteristics of quadratic functions including axis of symmetry, vertex, min/max, y- intercept, domain and range. (LT 7) In addition to graphing from standard form, we can also graph by performing. This works very well when the quadratic is written in vertex form. Standard Form y = ax 2 + bx + c b 2a Vertex Form y = a(x h) 2 + k y = x! 4x + 4 y = (x 2)! h y = x! + 6x + 8 y = (x + 3)! 1 y = 3x! 12x 8 y = 3(x + 2)! + 4 y = 2x! + 12x + 19 y = 2(x + 3)! + 1 Compare the values of!!! Vertex Form of a Quadratic Equation: and h in each row. Write a formula to show the relationship between!!! and h. The vertex is and the axis of symmetry line is Graph the equations on your calculator (Use a window of - 7 to 7.) Find the vertex of each. y=(x- 2) y=(x- 3) 2-5 y=(x+4) 2-3 Now let s transform the parent function y = x 2. Go to and click Start Graphing. In the first box, type the parent function. You can use the carrot button (^) for square. You should see a red parabola appear on the graph. In the second box, write the following equation: y = ax!. Click the blue button that appears to make a slider. Experiment with the slider, then fill in the blanks below When a is negative, the parabola (this is called a reflection) When a > 0, the parabola When 0 < a < 1, the parabola 15

16 Add parenthesis and the letter h to Box 2 so that it says this: y = a(x h)!. When h > 0, the parabola shifts When h < 0, the parabola shifts Now go back to Box 2 and add + k so that Box 1 says this: y = a(x h)! + k. When k > 0, the parabola shifts When k < 0, the parabola shifts Example 1: Describe the transformations applied to the parent quadratic function. Then determine the vertex, equation of the axis of symmetry, and whether the parabola opens up or down (Has a min or max). a.! y = (x 4)2 5 b.! y = 2(x +3)2 +6 c.! y = 1 2 x 2 4 Transformation: Equation of AOS: Vertex: Min or Max Example2: Graph! y = 2(x 4) Determine how far up/down/right/left, reflections and the vertex shifts, then graph it. 2. State the equation of the axis of symmetry and graph. 2. Determine if the parabola opens up or down (min or max). 4. When a=1, the points are graphed like this from the vertex: over- 1- up- 1, over- 2- up- 4, over- 3- up- 9. If a is not 1 then we have a vertical stretch of a. Multiply the up part of this pattern by a. Then graph three points on either side of the parabola. 5. Connect the points with a smooth curve. 16

17 Example 3: Graph! y = 1 2 (x +2)2 4 Example 4: Graph y = 2 x 3! + 9 Example 5: Graph y = 3(x +4) 2 +2 Example 6: Graph y =.5 x + 1! 5 17

18 Example 7: Graph y =(x +2) 2 Example 8: Graph y = x 1! + 4 Example 9: Graph y =3(x +4) 2 6 Example 10: Graph y =!! x! 3 18

19 Writing Quadratic Functions Date: After this lesson and practice, I will be able to rewrite quadratic equations from standard to vertex and vice versa. (LT 8) can write quadratic equations given a graph or given a vertex and the y-intercept (without a calculator).. (LT 9) write quadratic expressions/functions/equations given the roots/zeros/x- intercepts/solutions. (LT 10) We have explored several properties of quadratic equations written in form and form. Let s discuss the advantages of each equation form: Advantages to Standard Form Advantages to Vertex Form Rewriting Quadratic Equations (LT 8) Since each equation form has advantages, it is important to be able to rewrite equations in multiple forms. Example 1: Rewrite the vertex form equation in standard form and vice versa. a.! y = 2x 2 +10x +7 b.! y = 3(x +2)2 6 c. y = x 2 + 2x + 4 d. y = 3x 2 12x +17 e. y = 1 ( 2 x 4 ) f. y = 2( x + 5)

20 Writing Quadratic Equations from a Graph (LT 9) Another important skill is being able to write quadratic equations when given the graph or points. Example 2: Write the equation for each parabola in vertex form. a. b. c. Example 3: Write the equation of the parabola with the given vertex, y- intercept or point. Write in vertex form then rewrite in standard form. a. vertex at (- 3, 6) and a y- intercept of 33. Vertex form: standard form: b.. vertex (- 2,- 1) and the graph contains the point (4,11) Vertex form: standard form: c. vertex: (50,1000) and the y- int. is Vertex form: standard form: 20

21 We can also write the equation of quadratics from the solutions (or roots or zeros or intercepts!). This should look familiar we did it at the end of the last unit! Remember from last unit: Write an equation with solutions of - 7 and 3. Compare y 1 = x 2 + 4x 21 and 0 = x 2 + 4x 21 Let s compare some vocabulary: Term Definition Written as Answered as Solution Value(s) of x that make the equations true. These can be real or complex. x = - 7 and 3 or {- 7, 3} Equations = 0 x 2 + 4x 21 = 0 Root Value(s) of the variable that cause the polynomial to evaluate to zero. These can be real or complex. x = - 7 and 3 Polynomial Expression x 2 + 4x 21 Does NOT have a vertex, min or max. Does NOT have a vertex, min or max. Zero Value(s) in the argument for which the function is zero. These can be real or complex. x = - 7 and 3 Function as f(x) or y= f(x) = x 2 + 4x 21 or y = x 2 + 4x 21 Has a y- intercept, min or max and vertex. x- intercept An ordered pair on the graph that has a y value of zero. These must be real. (- 7,0) and (3,0) Function as f(x) or y= f(x) = x 2 + 4x 21 or y = x 2 + 4x 21 Has a y- intercept, min or max and vertex. Forms of Quadratics: Expression ax 2 +bx+c Standard Form ax 2 +bx+c=0 y=ax 2 +bx+c x- intercept Form y=(x- a)(x- b) Factored form: y=(x- a)(x- b) or 0=(x- a)(x- b) Vertex Form y=a(x- h) 2 +k 21

22 Given The Roots/Zeros/Intercepts/Solutions (LT 10) Example 1: Write a quadratic function in x- intercept/factored form with the following zeros: (Since this asks for zeros and y- intercepts the equation should be y = or f(x) = ) A) 2 and - 4 B) 4 and 4 C) 0 and - 7 Graph the equation from Example 1A in your graphing calculator or on Desmos. What point appears to be the vertex of this equation? How does the x- coordinate of the vertex relate to the two x- intercepts? Example 2: Write a quadratic equation in x- intercept or factored form with the following zeros. Then find the vertex of each equation. (Since this asks for zeros and x- intercept and asks you to find vertex the equation should be y = or f(x) =) A) - 1 and 3 *B) - 4i and 4i C) 0 and 8 Example 3: Write a quadratic equation in standard form that has the following solutions. (Since this asks for solutions ax 2 +bx+c=0. This is really from the another unit.) A. - 3i and 3i B. 0 and 2 C. 3 and - 5 Example 4: Write a quadratic equation in standard form that has the following x- intercepts. (Since it says x- intercepts standard must =y so y=ax 2 +bx+c) A. (0,0) a,d (- 3,0) B. (5,0) and (- 2,0) 22

23 Return of Completing the Square! Date: After this lesson and practice, I will be able to write quadratic equations in vertex form by completing the square. (LT 11) In the last unit, we completed the square as a way to solve quadratic equations. Now we will use the same strategy to rewrite equations into vertex form. Warm Up: Solve this equation by Completing the Square: 0 = x 2 10x + 22 Now Rewrite y = x 2 10x + 22 in vertex form using completing the square That form of quadratic equation should look familiar what form does it look like? So in addition to being a nice way to solve quadratics, Completing the Square is also a nice way to convert Standard Form equations to Vertex form. Let s practice! Write y = x 2 + 6x + 2in vertex form by completing the square, and state the vertex. How could you verify your answer? Example 5: Rewrite each equation into vertex form then state the vertex. a) y = x 2 10x 2 b) y = x 2 +5x + 3 c) y = x 2 4x +5 23

24 Ifa 1, then first divide it out of the x 2 and x terms. d) y = 3x 2 6x + 4 e) y = 2x 2 16x 25 f) y = 2x 2 + 3x + 2 Example 6: Write each equation in vertex form by completing the square. Identify the vertex. A) y = 2x 2 8x +1 B) y = x 2 + 4x 7 C) y= x x Example 7: The function P(s) = s s 2000 models the monthly profit P from selling sweaters at price s. a) Use completing the square to rewrite the function in vertex form. P(s) = b) Determine the maximum monthly profit, and the price at which the sweaters should be sold to attain that maximum. Sell at $ and earn $ max profit. c) Describe the practical domain and range for the profit function. Domain: Range: 24

25 Return of Real World Problems! Date: After this lesson and practice, I will be able to apply quadratics functions to real life situations without using the graphing calculator. (LT 4R) In previous lessons we have used our calculator to answer questions about real life situations. Now lets do this with what we have recently learned. Example 1: The income from ticket sales for a concert is modeled by the function I(p) = -50p p, where p is the price of a ticket. a. Calculate the maximum value of the function. (In other words, how high does the income go?) b. What price should be charged in order to attain the maximum income? Do part a and b again, but this time, use the graph in your calculator. Example 2: A pair of numbers has a sum of 8. Find their maximum product. Equation: Example 3: A rectangle has a perimeter of 24 inches. What dimensions would maximize the area? A(x) = 25

26 Example 4: A rectangle has the dimensions w What width will maximize the area? What is the maximum area? ( ) and ( 260 2w ) in feet. A(w) = Example 5: A company knows that 2.5p +500 models the number of unicycles it sells per month, where p is the price of a unicycle. Revenue from sales is the times the. What price will maximize revenue? What is the maximum revenue? R(p) = Example 6: An apartment complex has 100 two- bedroom units. The monthly profit in dollars realized from renting out the apartments is given by: P(x) = - 10x x 50,000 a. How many units must be rented out to maximize profits? b. What is the maximum profit realizable? c. How many apartments must be rented before a profit is made? 26

27 Example 7 : A bakery knows from market research that - 5p models the of number of pies sold on average day, where p is the price of a pie in dollars. (Hint: make into Revenue by ) a. Write an equation to represent the revenue the bakery can expect on an average day. R(p) = b. What price will maximize the revenue? c. What is the maximum revenue? Example 8: Profit earned from sponsoring a basketball game between teachers and students is modeled by y = 100x x 650, where x is the price of a ticket. Graph the function on your calculator, make a sketch below that includes the important information, and answer the following questions. a. What is the domain of the function? b. What profit is earned if the price of the ticket is $5? c. What price will lead to max profit? What is the max profit? Example 9: Smoke jumpers are firefighters who parachute into areas near forest fires. A jumper s height during free fall (before the parachute opens) is modeled by: h(t) = 16t 2 + height of plane where t is time in seconds. If a jumper s plane is 1400 ft. off the ground when he jumps, how long is he in free fall before his parachute opens at 1000 ft.? h(t) = Example 10 If Jimmy Fallon dropped a T.V. from a height of 1000 feet, how long would it take to hit the ground? How long would it take to land on a building 20 feet tall. Use the formula: h(t) = - 16t 2 + s h(t) = h = height when the object landed. If it landed on the ground, then h =. If it landed on the top of a 20 ft. building, then h =. t = the time it takes for the object to fall. s = height from which the object is dropped or. where the object started! time to ground time to building 27

28 Unit I can use the discriminant to determine the number and type of solutions/zeros. As we ve seen, quadratic equations can have or solutions. Let s discover how to quickly determine which type of solution a given quadratic equation has Discriminant: Given 2 ax + bx + c = 0, the discriminant is. The discriminant does not include the radical symbol! Fill out the table below. Equation A) 3x2 5x = 2 Value of Discriminant b 2 4ac Number (0, 1 or 2) and Type of Solutions (Real or Complex) B) 2x2 +8x = 12 C) 9x2 +12x + 4 = 0 D) 2 4x 8x+ 1= 0 E) 2x2 =7x 8 Summary: Value of Discriminant Number and Types of Solutions Positive Zero Negative 28

29 Example 2: Find the discriminant and give the number and type of solutions. Do not solve A) x + 10x+ 23 = 0 B) x + 10x+ 25 = 0 C) x + 10x+ 27 = 0 0 real, 2 complex 1 real 2 real 0 real, 2 complex 1 real 2 real 0 real, 2 complex 1 real 2 real One more use for the discriminant! If the discriminant is a perfect square, the quadratic expression is factorable under rational numbers. If it is not a perfect square, the quadratic expression is prime under rational numbers. 29