Converting Quadratic Equations Worksheet - Coach Forrester - Free Printable

Let’s solve each problem step by step. We’ll convert between vertex form and standard form of quadratic equations.

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Part 1: Convert from Vertex Form to Standard Form

Vertex form is:
> y = a(x - h)² + k

Standard form is:
> y = ax² + bx + c

We expand the squared term, distribute, and combine like terms.

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Problem 1:
y = -(x - 1)² - 1

Step 1: Expand (x - 1)² → x² - 2x + 1
Step 2: Multiply by -1 → -x² + 2x - 1
Step 3: Subtract 1 → -x² + 2x - 1 - 1 = -x² + 2x - 2

✔ Final Answer for #1: y = -x² + 2x - 2

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Problem 2:
y = 2(x - 2)² - 3

Step 1: Expand (x - 2)² → x² - 4x + 4
Step 2: Multiply by 2 → 2x² - 8x + 8
Step 3: Subtract 3 → 2x² - 8x + 8 - 3 = 2x² - 8x + 5

✔ Final Answer for #2: y = 2x² - 8x + 5

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Problem 3:
y = (x + 4)² + 4

Note: (x + 4)² = (x - (-4))² — same idea.

Step 1: Expand (x + 4)² → x² + 8x + 16
Step 2: Add 4 → x² + 8x + 16 + 4 = x² + 8x + 20

✔ Final Answer for #3: y = x² + 8x + 20

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Part 2: Convert from Standard Form to Vertex Form

Standard form: y = ax² + bx + c
Vertex form: y = a(x - h)² + k

We use “completing the square”.

Steps:
1. Group x² and x terms.
2. Factor out ‘a’ if needed (if a ≠ 1).
3. Take half of b, square it, add and subtract inside the parentheses.
4. Rewrite as perfect square trinomial.
5. Simplify constants outside.

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Problem 4:
y = x² - 8x + 15

Step 1: Focus on x² - 8x
Half of -8 is -4 → square it: 16
Add and subtract 16:

y = (x² - 8x + 16 - 16) + 15
= (x² - 8x + 16) - 16 + 15
= (x - 4)² - 1

✔ Final Answer for #4: y = (x - 4)² - 1

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Problem 5:
y = x² - 4x

Step 1: Half of -4 is -2 → square it: 4
Add and subtract 4:

y = (x² - 4x + 4 - 4)
= (x - 2)² - 4

✔ Final Answer for #5: y = (x - 2)² - 4

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Problem 6:
y = x² + 8x + 18

Step 1: Half of 8 is 4 → square it: 16
Add and subtract 16:

y = (x² + 8x + 16 - 16) + 18
= (x + 4)² + 2

✔ Final Answer for #6: y = (x + 4)² + 2

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Problem 7:
y = x² + 6x + 3

Step 1: Half of 6 is 3 → square it: 9
Add and subtract 9:

y = (x² + 6x + 9 - 9) + 3
= (x + 3)² - 6

✔ Final Answer for #7: y = (x + 3)² - 6

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Problem 8:
y = x² - 2x + 5

Step 1: Half of -2 is -1 → square it: 1
Add and subtract 1:

y = (x² - 2x + 1 - 1) + 5
= (x - 1)² + 4

✔ Final Answer for #8: y = (x - 1)² + 4

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Problem 9:
y = x² - 8x + 17

Step 1: Half of -8 is -4 → square it: 16
Add and subtract 16:

y = (x² - 8x + 16 - 16) + 17
= (x - 4)² + 1

✔ Final Answer for #9: y = (x - 4)² + 1

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Final Answer:
1) y = -x² + 2x - 2
2) y = 2x² - 8x + 5
3) y = x² + 8x + 20
4) y = (x - 4)² - 1
5) y = (x - 2)² - 4
6) y = (x + 4)² + 2
7) y = (x + 3)² - 6
8) y = (x - 1)² + 4
9) y = (x - 4)² + 1