Factoring Quadratic Equations - Free Printable

Let’s solve each quadratic equation step by step. We’ll factor them and then find the values of the variable that make each equation true.

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1) 2x² + 3x + 1 = 0

We look for two numbers that multiply to (2 × 1) = 2 and add to 3 → those are 2 and 1.

Split middle term:
2x² + 2x + x + 1 = 0
Group: (2x² + 2x) + (x + 1) = 0
Factor: 2x(x + 1) + 1(x + 1) = 0
→ (2x + 1)(x + 1) = 0

Set each factor to zero:
2x + 1 = 0 → x = -½
x + 1 = 0 → x = -1

✔ Final: x = -½ or -1

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2) 2y² + 7y + 3 = 0

Multiply 2 × 3 = 6, find two numbers that multiply to 6 and add to 7 → 6 and 1.

Split: 2y² + 6y + y + 3 = 0
Group: (2y² + 6y) + (y + 3) = 0
Factor: 2y(y + 3) + 1(y + 3) = 0
→ (2y + 1)(y + 3) = 0

Solutions:
2y + 1 = 0 → y = -½
y + 3 = 0 → y = -3

✔ Final: y = -½ or -3

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3) 2z² – 8z + 8 = 0

First, factor out 2:
2(z² – 4z + 4) = 0
Now factor inside: z² – 4z + 4 = (z – 2)²
So: 2(z – 2)² = 0 → (z – 2) = 0 → z = 2

But let’s check the given answer format: they wrote (2z – 4)(z – 2) = 0
That’s also correct because 2z – 4 = 2(z – 2), so it’s same as above.

Set factors to zero:
2z – 4 = 0 → z = 2
z – 2 = 0 → z = 2

Only one solution: z = 2

✔ Final: z = 2

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4) 2a² – 3a – 5 = 0

Multiply 2 × (-5) = -10. Find two numbers that multiply to -10 and add to -3 → -5 and 2.

Split: 2a² – 5a + 2a – 5 = 0
Group: (2a² – 5a) + (2a – 5) = 0
Factor: a(2a – 5) + 1(2a – 5) = 0
→ (2a – 5)(a + 1) = 0

Solutions:
2a – 5 = 0 → a = 5/2
a + 1 = 0 → a = -1

✔ Final: a = 5/2 or -1

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5) 2b² – 5b + 3 = 0

Multiply 2 × 3 = 6. Numbers that multiply to 6 and add to -5 → -3 and -2.

Split: 2b² – 3b – 2b + 3 = 0
Group: (2b² – 3b) + (-2b + 3) = 0
Factor: b(2b – 3) –1(2b – 3) = 0
→ (2b – 3)(b – 1) = 0

Solutions:
2b – 3 = 0 → b = 3/2
b – 1 = 0 → b = 1

✔ Final: b = 3/2 or 1

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6) 2c² + 7c – 4 = 0

Multiply 2 × (-4) = -8. Numbers that multiply to -8 and add to 7 → 8 and -1.

Split: 2c² + 8c – c – 4 = 0
Group: (2c² + 8c) + (-c – 4) = 0
Factor: 2c(c + 4) –1(c + 4) = 0
→ (2c – 1)(c + 4) = 0

Solutions:
2c – 1 = 0 → c = 1/2
c + 4 = 0 → c = -4

✔ Final: c = 1/2 or -4

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7) 2d² – 9d + 9 = 0

Multiply 2 × 9 = 18. Numbers that multiply to 18 and add to -9 → -6 and -3.

Split: 2d² – 6d – 3d + 9 = 0
Group: (2d² – 6d) + (-3d + 9) = 0
Factor: 2d(d – 3) –3(d – 3) = 0
→ (2d – 3)(d – 3) = 0

Solutions:
2d – 3 = 0 → d = 3/2
d – 3 = 0 → d = 3

✔ Final: d = 3/2 or 3

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8) 2e² – 3e + 1 = 0

Multiply 2 × 1 = 2. Numbers that multiply to 2 and add to -3 → -2 and -1.

Split: 2e² – 2e – e + 1 = 0
Group: (2e² – 2e) + (-e + 1) = 0
Factor: 2e(e – 1) –1(e – 1) = 0
→ (2e – 1)(e – 1) = 0

Solutions:
2e – 1 = 0 → e = 1/2
e – 1 = 0 → e = 1

✔ Final: e = 1/2 or 1

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9) 2f² – 6f – 8 = 0

First, factor out 2:
2(f² – 3f – 4) = 0
Now factor: f² – 3f – 4 = (f – 4)(f + 1)
So: 2(f – 4)(f + 1) = 0 → same as (2f + 2)(f – 4)? Let’s check:

Given answer: (2f + 2)(f – 4) = 0 → expand: 2f·f = 2f², 2f·(-4)= -8f, 2·f=2f, 2·(-4)=-8 → total: 2f² -6f -8 ✔️

Set to zero:
2f + 2 = 0 → f = -1
f – 4 = 0 → f = 4

✔ Final: f = -1 or 4

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10) 2g² – g – 10 = 0

Multiply 2 × (-10) = -20. Numbers that multiply to -20 and add to -1 → -5 and 4.

Split: 2g² – 5g + 4g – 10 = 0
Group: (2g² – 5g) + (4g – 10) = 0
Factor: g(2g – 5) + 2(2g – 5) = 0
→ (2g – 5)(g + 2) = 0

Solutions:
2g – 5 = 0 → g = 5/2
g + 2 = 0 → g = -2

✔ Final: g = 5/2 or -2

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11) 2h² + 9h + 7 = 0

Multiply 2 × 7 = 14. Numbers that multiply to 14 and add to 9 → 7 and 2.

Split: 2h² + 7h + 2h + 7 = 0
Group: (2h² + 7h) + (2h + 7) = 0
Factor: h(2h + 7) + 1(2h + 7) = 0
→ (2h + 7)(h + 1) = 0

Solutions:
2h + 7 = 0 → h = -7/2
h + 1 = 0 → h = -1

✔ Final: h = -7/2 or -1

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12) 2i² – i – 3 = 0

Multiply 2 × (-3) = -6. Numbers that multiply to -6 and add to -1 → -3 and 2.

Split: 2i² – 3i + 2i – 3 = 0
Group: (2i² – 3i) + (2i – 3) = 0
Factor: i(2i – 3) + 1(2i – 3) = 0
→ (2i – 3)(i + 1) = 0

Solutions:
2i – 3 = 0 → i = 3/2
i + 1 = 0 → i = -1

✔ Final: i = 3/2 or -1

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13) 2j² – 13j + 6 = 0

Multiply 2 × 6 = 12. Numbers that multiply to 12 and add to -13 → -12 and -1.

Split: 2j² – 12j – j + 6 = 0
Group: (2j² – 12j) + (-j + 6) = 0
Factor: 2j(j – 6) –1(j – 6) = 0
→ (2j – 1)(j – 6) = 0

Solutions:
2j – 1 = 0 → j = 1/2
j – 6 = 0 → j = 6

✔ Final: j = 1/2 or 6

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14) 2k² + 3k – 9 = 0

Multiply 2 × (-9) = -18. Numbers that multiply to -18 and add to 3 → 6 and -3.

Split: 2k² + 6k – 3k – 9 = 0
Group: (2k² + 6k) + (-3k – 9) = 0
Factor: 2k(k + 3) –3(k + 3) = 0
→ (2k – 3)(k + 3) = 0

Solutions:
2k – 3 = 0 → k = 3/2
k + 3 = 0 → k = -3

✔ Final: k = 3/2 or -3

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15) 2m² + 7m – 15 = 0

Multiply 2 × (-15) = -30. Numbers that multiply to -30 and add to 7 → 10 and -3.

Split: 2m² + 10m – 3m – 15 = 0
Group: (2m² + 10m) + (-3m – 15) = 0
Factor: 2m(m + 5) –3(m + 5) = 0
→ (2m – 3)(m + 5) = 0

Solutions:
2m – 3 = 0 → m = 3/2
m + 5 = 0 → m = -5

✔ Final: m = 3/2 or -5

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16) 2n² – 8 = 0

This is a difference of squares? Not quite — but we can factor out 2 first:

2(n² – 4) = 0 → n² – 4 = (n – 2)(n + 2)

So: 2(n – 2)(n + 2) = 0 → same as (2n – 4)(n + 2) = 0? Let’s check:

(2n – 4)(n + 2) = 2n·n = 2n², 2n·2=4n, -4·n=-4n, -4·2=-8 → 2n² + 0n -8 ✔️

Set to zero:
2n – 4 = 0 → n = 2
n + 2 = 0 → n = -2

✔ Final: n = 2 or -2

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Final Answer:
1) x = -½ or -1
2) y = -½ or -3
3) z = 2
4) a = 5/2 or -1
5) b = 3/2 or 1
6) c = 1/2 or -4
7) d = 3/2 or 3
8) e = 1/2 or 1
9) f = -1 or 4
10) g = 5/2 or -2
11) h = -7/2 or -1
12) i = 3/2 or -1
13) j = 1/2 or 6
14) k = 3/2 or -3
15) m = 3/2 or -5
16) n = 2 or -2