Math worksheet for practicing converting quadratic equations to vertex form.

Let’s solve each problem step by step. We’re converting quadratic equations from standard form (ax² + bx + c) to vertex form:
y = a(x – h)² + k, where (h, k) is the vertex.

We’ll use completing the square for most of these. Let’s go one by one.

---

Problem 1: y = x² + 4x + 3

Step 1: Group x terms:
y = (x² + 4x) + 3

Step 2: Complete the square inside the parentheses.
Take half of 4 → 2, then square it → 4.
Add and subtract 4 inside:

y = (x² + 4x + 4 - 4) + 3
= (x² + 4x + 4) - 4 + 3
= (x + 2)² - 1

✔ Final Answer: y = (x + 2)² - 1

---

Problem 2: y = x² - 6x + 8

Group: y = (x² - 6x) + 8
Half of -6 is -3, square it → 9

y = (x² - 6x + 9 - 9) + 8
= (x - 3)² - 9 + 8
= (x - 3)² - 1

✔ Final Answer: y = (x - 3)² - 1

---

Problem 3: y = x² + 10x + 25

Wait — this is already a perfect square!

x² + 10x + 25 = (x + 5)²

So y = (x + 5)² + 0

✔ Final Answer: y = (x + 5)²

---

Problem 4: y = 2x² + 8x + 5

Factor out the 2 first from the x terms:

y = 2(x² + 4x) + 5

Complete the square inside: half of 4 is 2, square is 4.

But we can’t just add 4 — we have to account for the 2 outside.

So:
y = 2(x² + 4x + 4 - 4) + 5
= 2[(x + 2)² - 4] + 5
= 2(x + 2)² - 8 + 5
= 2(x + 2)² - 3

✔ Final Answer: y = 2(x + 2)² - 3

---

Problem 5: y = -x² + 2x + 7

Factor out -1 from x terms:

y = -(x² - 2x) + 7

Complete the square: half of -2 is -1, square is 1.

y = -(x² - 2x + 1 - 1) + 7
= -[(x - 1)² - 1] + 7
= -(x - 1)² + 1 + 7
= -(x - 1)² + 8

✔ Final Answer: y = -(x - 1)² + 8

---

Problem 6: y = 3x² - 12x + 10

Factor out 3:

y = 3(x² - 4x) + 10

Half of -4 is -2, square is 4.

y = 3(x² - 4x + 4 - 4) + 10
= 3[(x - 2)² - 4] + 10
= 3(x - 2)² - 12 + 10
= 3(x - 2)² - 2

✔ Final Answer: y = 3(x - 2)² - 2

---

Problem 7: y = -2x² + 8x - 5

Factor out -2:

y = -2(x² - 4x) - 5

Half of -4 is -2, square is 4.

y = -2(x² - 4x + 4 - 4) - 5
= -2[(x - 2)² - 4] - 5
= -2(x - 2)² + 8 - 5
= -2(x - 2)² + 3

✔ Final Answer: y = -2(x - 2)² + 3

---

Problem 8: y = ½x² + 3x + 1

Factor out ½:

y = ½(x² + 6x) + 1

Half of 6 is 3, square is 9.

y = ½(x² + 6x + 9 - 9) + 1
= ½[(x + 3)² - 9] + 1
= ½(x + 3)² - 9/2 + 1
= ½(x + 3)² - 7/2

✔ Final Answer: y = ½(x + 3)² - 7/2

---

Problem 9: y = -¼x² + x - 2

Factor out -¼:

y = -¼(x² - 4x) - 2

Half of -4 is -2, square is 4.

y = -¼(x² - 4x + 4 - 4) - 2
= -¼[(x - 2)² - 4] - 2
= -¼(x - 2)² + 1 - 2
= -¼(x - 2)² - 1

✔ Final Answer: y = -¼(x - 2)² - 1

---

Problem 10: y = 5x² - 20x + 18

Factor out 5:

y = 5(x² - 4x) + 18

Half of -4 is -2, square is 4.

y = 5(x² - 4x + 4 - 4) + 18
= 5[(x - 2)² - 4] + 18
= 5(x - 2)² - 20 + 18
= 5(x - 2)² - 2

✔ Final Answer: y = 5(x - 2)² - 2

---

Problem 11: y = -3x² + 12x - 10

Factor out -3:

y = -3(x² - 4x) - 10

Half of -4 is -2, square is 4.

y = -3(x² - 4x + 4 - 4) - 10
= -3[(x - 2)² - 4] - 10
= -3(x - 2)² + 12 - 10
= -3(x - 2)² + 2

✔ Final Answer: y = -3(x - 2)² + 2

---

Problem 12: y = 4x² + 16x + 15

Factor out 4:

y = 4(x² + 4x) + 15

Half of 4 is 2, square is 4.

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

✔ Final Answer: y = 4(x + 2)² - 1

---

## ✔ Final Answers:

1. y = (x + 2)² - 1
2. y = (x - 3)² - 1
3. y = (x + 5)²
4. y = 2(x + 2)² - 3
5. y = -(x - 1)² + 8
6. y = 3(x - 2)² - 2
7. y = -2(x - 2)² + 3
8. y = ½(x + 3)² - 7/2
9. y = -¼(x - 2)² - 1
10. y = 5(x - 2)² - 2
11. y = -3(x - 2)² + 2
12. y = 4(x + 2)² - 1