Solving Quadratics Notes Packet

Name____________________________________________ Period________ Date_______________________________ Algebra 1 Solving Quadratics Notes Packet Fill in the table of values for each equation below. Then, graph the points. 1)

𝑦𝑦 = 𝑥𝑥 2 + 𝑥𝑥 − 2

At which coordinate points does your graph intersect the x-axis?

2)

𝑦𝑦 = 𝑥𝑥 2 − 2𝑥𝑥 − 3

At which coordinate points does your graph intersect the x-axis?

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Name____________________________________________ Period________ Date_______________________________ 3)

𝑦𝑦 = 𝑥𝑥 2 − 4

At which coordinate points does your graph intersect the x-axis?

Now go back and factor the right side of each equation. 1a)

2a)

3a)

What do you notice?

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Name____________________________________________ Period________ Date_______________________________ SOLVING QUADRATICS What does it mean to SOLVE a quadratic equation? __________________________________________________________________________________________________ How many solutions does each example have? a)

b)

c)

How can we tell the number of solutions if we have a graph? __________________________________________________________________________________________________ How can we tell the number of solutions if we only have an equation? __________________________________________________________________________________________________ METHOD 1: SOLVING QUADRATICS BY FACTORING When we solve a quadratic equation by factoring, we use the following steps: Steps Step 1:

𝒚𝒚 = 𝒙𝒙𝟐𝟐 − 𝟓𝟓𝟓𝟓 + 𝟒𝟒

Step 2:

Step 3:

Step 4:

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Name____________________________________________ Period________ Date_______________________________ PRACTICE EXAMPLES Solve each equation using the steps from the previous page. SHOW YOUR WORK for each step. EXAMPLE 1

2

𝑦𝑦 = 𝑥𝑥 + 5𝑥𝑥 + 6

EXAMPLE 2

Step 1:

Step 1:

Step 2:

Step 2:

Step 3:

Step 3:

Step 4:

Step 4:

EXAMPLE 3

𝑦𝑦 = 𝑥𝑥 2 − 8𝑥𝑥

EXAMPLE 4

Step 1:

Step 1:

Step 2:

Step 2:

Step 3:

Step 3:

Step 4:

Step 4:

𝑦𝑦 = 𝑥𝑥 2 + 3𝑥𝑥 − 2

𝑦𝑦 = 𝑥𝑥 2 − 25

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Name____________________________________________ Period________ Date_______________________________ HOMEWORK Based on the examples on pages 3 and 4, what MUST you be able to do in order to solve a quadratic equation using factoring? __________________________________________________________________________________________________ __________________________________________________________________________________________________ Why do you think this is? You may need to graph each equation to answer this question. __________________________________________________________________________________________________ __________________________________________________________________________________________________

PRACTICE Solve each equation by factoring. 1)

𝑦𝑦 = 𝑥𝑥 2 + 4𝑥𝑥 + 3

2)

𝑦𝑦 = 𝑥𝑥 2 − 2𝑥𝑥

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Name____________________________________________ Period________ Date_______________________________ 3) 𝑦𝑦 = 𝑥𝑥 2 − 121 4) 𝑦𝑦 = 𝑥𝑥 2 + 6𝑥𝑥 − 27

5)

𝑦𝑦 = 𝑥𝑥 2 + 11𝑥𝑥 + 24

6)

𝑦𝑦 = 𝑥𝑥 2 − 64

7)

𝑦𝑦 = 2𝑥𝑥 2 + 10𝑥𝑥

8)

𝑦𝑦 = 𝑥𝑥 2 + 𝑥𝑥 − 2

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Name____________________________________________ Period________ Date_______________________________ METHOD 2: SOLVING QUADRATICS WITH THE QUADRATIC FORMULA We can solve ANY quadratic equation using the QUADRATIC FORMULA. When we are not able to factor to solve, we can use the quadratic formula.

Quadratic Formula:

USING THE QUADRATIC FORMULA Step 1:

STEPS

𝒚𝒚 = 𝒙𝒙𝟐𝟐 + 𝟑𝟑𝟑𝟑 − 𝟐𝟐

Step 2:

Step 3:

Step 4:

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Name____________________________________________ Period________ Date_______________________________ CLASSWORK Solve each equation using quadratic formula. SHOW YOUR WORK for each step. EXAMPLE 1

2

𝑦𝑦 = 2𝑥𝑥 − 2𝑥𝑥 − 5

EXAMPLE 2

Step 1:

Step 1:

Step 2:

Step 2:

Step 3:

Step 3:

Step 4:

Step 4:

EXAMPLE 3

2

𝑦𝑦 = 3𝑥𝑥 − 10

EXAMPLE 4

Step 1:

Step 1:

Step 2:

Step 2:

Step 3:

Step 3:

Step 4:

Step 4:

𝑦𝑦 = 𝑥𝑥 2 + 5𝑥𝑥

𝑦𝑦 = 𝑥𝑥 2 + 3𝑥𝑥 − 4

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Name____________________________________________ Period________ Date_______________________________ HOMEWORK Solve each equations using the quadratic formula. SHOW ALL OF YOUR WORK. 1)

𝑦𝑦 = 𝑥𝑥 2 + 10𝑥𝑥 − 2

2)

𝑦𝑦 = 𝑥𝑥 2 − 8𝑥𝑥 − 20

3)

𝑦𝑦 = −3𝑥𝑥 2 + 11𝑥𝑥

4)

𝑦𝑦 = 2𝑥𝑥 2 − 7

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Name____________________________________________ Period________ Date_______________________________ 5) 𝑦𝑦 = 4𝑥𝑥 2 − 6𝑥𝑥 − 1 6) 𝑦𝑦 = −𝑥𝑥 2 + 7𝑥𝑥 − 18

7)

𝑦𝑦 = 4𝑥𝑥 2 − 25

8)

𝑦𝑦 = −5𝑥𝑥 2 − 18𝑥𝑥

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