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

Using the Quadratic FormulaKey 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.

The quadratic formula is a formula that can be used to find the roots of a quadratic equation. Because of that, it can be used to solve any quadratic equation. Part of the quadratic is the discriminant, which tells the number of real roots the equation will have.

Vocabulary: Quadratic, roots, zeroes, solutions, Quadratic formula, discriminant, Real root, complex root, radical, radicand

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Part 1: Activation of Prior Knowledge

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Both equations below cannot be factored. Of the two equations, which is easier to solve using completing the square? Why?

Equation A

x2+10x+8=0

Equation B

x2−7 x+3=0

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Part 2: Guided Practice

The ________________________ is used to determine the _______________ real solutions of a quadratic equation.

The quadratic formula is used to determine the actual ________________ .

To use the quadratic formula, we must have a quadratic equation in _____________________________ so that we can identify a, b, and c.

x=−b±√b2−4ac2a

Example 1: Use the quadratic formula to find the solutions. Be sure to simplify the radical fully.

y=x2+2 x−1

Example 2: Use the graph to estimate the solutions, then the quadratic formula to find the solutions: y=x2+5 x+8

Estimate: ________________

Solve using quadratic formula:3

Advantages of the Quadratic Formula Disadvantages of the Quadratic Formula

Part 3: Independent Practice (MILD)

1. Use the graph to estimate the solutions of the equation below, then use the quadratic formula to find the exact solutions:

y=x2−10 x+25

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Estimate: ________________

2. Which equation has the graph shown to the right?

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3. Estimate the solutions to the quadratic equation you picked in #2.

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4. Using the quadratic formula, find the solutions.

5. Use the quadratic formula to determine the solution(s) to the following equation: x2−5 x−14=0

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6. What is the solution set to the following equation 2m2−7m=13?

Part 4: Independent Practice (MEDIUM)

a) Factor and Solve (if possible) b) Use the Quadratic Formula to find the solutions.

1. y=x2−4 x−12

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2. y=−x2−6 x−9

3. y=2x2+ x−6

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4. y=x2+4

Part 5: Independent Practice (SPICY)

8. The area of a rectangle is 84 square meters. The length is 5 meters more than the width. Part A: Write an equation, using w to represent the width, that represents this scenario.

Part B: What is the length in meters? Solve algebraically. Remember to include units.

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Part C: What method did you use to solve the equation? Why?______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Mathletes

Natalya and Julia earn money by babysitting. Natalya charges an initial $2 fee and then $4 per hour from each family. Julia charges an initial $4 fee and then also charges $2 per hour. For what amount of hours will they charge the same amount for babysitting?

A. Write a system of equations and then solve it algebraically.

B. Graph to check your solution.

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“”I’m a mathlete

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

Using the Quadratic FormulaExit Ticket

Directions: Complete each problem by showing ALL work. Don’t forget to use MOLE! 1) Use the quadratic formula to determine the solutions of the equation 2 x2+10 x+7=0.

2) Complete the table below using the equation y=x2+12 x+36.a) Factor and solve. b) Determine the number of solutions using

the discriminant

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c) Solve using Quadratic Formula

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

Using the Quadratic FormulaHomework

Directions: Solve each problem. Show all work using MOLE. 1) Example 1: y=2x23 x−4 Example 2: y=x2−2 x−6Graphing:(Most efficient with whole number solutions and technology)

What is the number of solutions? Estimate or determine their values, if possible.

Can you factor this equation? Why or why not?

Solve using the Quadratic Formula or use the discriminant to show why it

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is not necessary.

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2) Solve the following equation by factoring and by using the quadratic formula.x2−7 x−8=0

Factoring Quadratic Formula

3) Which of the following is a solution to the system of equations below?

{ y=5 x−7−3 x−2 y=−12}

a) (-3, -2)b) (2, 3)c) (-2, -17)d) (2, -3)

4) How many real solutions does the following equation have −4 r2−4 r−6=0?

1) One real solution2) Two real solutions3) Infinite real solutions4) No real solutions

5) The area of a rectangle is x2+3x−18. If the length is p−3, what is the width?

1) p−62) p+63) x−94) x−3

6) The roots of a function are 73 and−7

2 . Which of the following could be the factors of the function?

1) (3 x+7 ) (2x+7 )2) ( x−7 ) (2 x+7 )3) (3 x−7 ) (2 x−7 )4) (3 x−1 ) (2x+7 )

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