Name Date ______________ HSA.REI.B.4.B Team

Using the discriminant to determine the number of real solutionsKey Takeaways:

Standard: Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

Quadratic equations can have zero, one, or two real roots. You can determine the number of roots that a quadratic has by finding the discriminant (d=b2−4 ac¿ and noticing:

If d<0, there are no real roots If d=0, there is one real root

If d>0, there are two real roots

Vocabulary: Quadratic, root, solution, real solution, discriminant, x-intercept____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Part 1: Activation of Prior Knowledge

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1. For each quadratic equation below, determine the number of real roots it has. Write the number of solutions under each graph.

A ¿ y=2 x2−x+4 B ¿ y=x2+2x+1C ¿ y=x2+5 x+4

2. Explain how you determined how many real roots each quadratic equation has.________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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Part 2: Guided Practice

The ____________________________ is a part of the quadratic formula that can be used to determine the number of ____________________________ a quadratic equation has.

Discriminant formula: _______________________

Example 1: Calculate the discriminant for each of the equations from the APK. Use your calculations to fill out the table below.

A ¿ y=2 x2−x+4 B ¿ y=x2+2x+1C ¿ y=x2+5 x+4

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Value of discriminant Number of real solutions the quadratic has

NegativeZeroPositive

Example 2: Determine the number of solutions to the equation below using the discriminant, then check the number of solutions by graphing on the coordinate plane.

y=x2+10 x+16=0

Example 3: How many real solutions does the quadratic equation below have?

y=x2 –10 x+25

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a) One real solutionb) Two real solutionsc) Three real solutionsd) No real solutions

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Part 3: Independent Practice (MILD)

1) How many real solutions does the quadratic equation below have? 2 x2−4 x+5=0

a) 1b) 2c) 3d) No real

2) Find the discriminant of the equation x2−5 x=0. a) 25b) 21c) 1d) 29

Write the letter of the graph that matches the value of the discriminant below.

3) b2−4 ac=2 _____________________4) b2−4 ac=0 _____________________5) b2−4 ac=−3 _______________________

6) What is the discriminant of the function below?

y=x2 –10 x+30

a) 20b) -20c) -220d) 0

7) For which value of c will −3 x2+6x+c=0 have only one real solution?

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a) c←3b) c=−3c) c>−3d) c=3

8) How many real solutions does x2−10 x+25=0 have?a) No solutionsb) One solutionc) Two solutionsd) Many solutions

9) Determine the number of solutions for each of the following quadratic equations. Use the discriminant to solve and then use your graphing calculator to check and make a sketch.

A) x2−1=0a) 0b) 1c) 2d) 3

B) x2−2 x+1=0a) 0b) 1c) 2d) 3

C) x2+3x+4=0a) 0b) 1c) 2d) 3

Part 4: Independent Practice (MEDIUM)

7) Determine the number of real roots for each equation in the table below. Then choose the statement below that is true about the number of real roots in each column.

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a. The total number of real roots in column A is greaterb. The total number of real roots in Column B is greaterc. The total number of real roots is equald. The relationship cannot be determined from the given information

8) Determine the number of real solutions using the discriminant and then use your graphing calculator to check and sketch the graph.

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a) y=x2−4 x−4# of real solutions _____________________

b) y=−x2−3# of real solutions ______________________

c) y=−x2−1x+6# of real solutions _____________________

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d ¿ y=x2−6 x+9# of real solutions _____________________

9) How is the discriminant useful? Be specific.

___________________________________________________________________________________________

___________________________________________________________________________________________

___________________________________________________________________________________________

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Part 5: Independent Practice (SPICY)

10. The length of a rectangle is 7 units more than its width. If the width is doubled and the length is increased by 2, the area is increased by 42 square units. Find the dimensions of the original rectangle.

11. Which statement is true about the systems of equations below?

2 x+8 y=6−5 x−20 y=−15

a. It has no solutionb. It has only one solutionc. It has two solutions d. It has infinite solutions

12. Ms. Blalock is selling tickets to a choral performance. On the first day of ticket sales the she sold 3 senior citizen tickets and 1 child ticket for a total of $38. She made $52 on the second day by selling 3 senior citizen tickets and 2 child tickets. How much more does a child’s ticket cost than a senior citizen ticket?

Mathletes

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“”I’m a mathlete

Name Date ______________ HSA.REI.B.4.B Team

Using the discriminant to determine the number of real solutionsExit Ticket

Directions: Complete each problem by showing ALL work. Don’t forget to use MOLE! 12

1. Determine the number of real solutions the equation below has by using the discriminant. Then, sketch the graph to confirm that your result is correct.

y=x2+8 x+15

2. What is the discriminant of the quadratic below?

f ( x )=x2−10 x+25

3. What does the discriminant that you calculated in #2 tell you about the number of real solutions that equation will have? Explain.________________________________________________________________________________________________________________________________________________________________________________________________________________________

Name Date ______________ HSA.REI.B.4.B Team

Using the discriminant to determine the number of real solutionsHomework

Directions: Solve each problem. Show all work using MOLE.

1) Use the equation y = x2 – 6x + 5 to complete parts a-d.a) Factor and Solve b) Use the discriminant to find the # of

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solutions.

c) Find the vertex. d) Graph the equation by using the solutions and the vertex and then label each one on your graph.

2) 3)

4) Which expression below is equivalent to

36b2−1

a) (18b−1 ) (18b+1 )b) (6b−1 ) (6b+1 )

5) How many roots does the graphed function shown have?

a) Oneb) Twoc) Noned) Three

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c) (18b+1 ) (18b+1 )d) (6b+1)(6b+1)

6) What is the rate of change for following table?

a) 0.2b) 0.5c) 5d) 8

7) Which shows the following expression factored completely?

6 x2+17x+7

A. (3 x+1)(2 x+7)B. (6 x+1)(1 x+7)C. (3 x+7)(2x+1)D. (6 x+7)(1 x+1)

8) What is the solution set to the following equation?k 2−10 k+26=8

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