Quadratic equations worksheet for 11 - Free Printable

Let’s solve each quadratic equation using the quadratic formula:

> For any quadratic equation in the form:
> ax² + bx + c = 0,
> the solutions are:
> x = [-b ± √(b² - 4ac)] / (2a)

We’ll also find the discriminant: D = b² - 4ac
- If D > 0 → two real, different roots
- If D = 0 → one real root (repeated)
- If D < 0 → no real roots (complex/imaginary)

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1. x² – 8x + 16 = 0

a=1, b=-8, c=16
D = (-8)² - 4(1)(16) = 64 - 64 = 0 → one real root
x = [8 ± √0]/2 = 8/2 = 4

✔ Root: x = 4 (repeated)

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2. 2n² – 16n + 40 = 0

Divide all by 2 to simplify: n² – 8n + 20 = 0
a=1, b=-8, c=20
D = 64 - 80 = -16 → no real roots

✔ Roots: Complex (not real)

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3. x² – 49 = 0

This is difference of squares: (x–7)(x+7)=0
Or use formula: a=1, b=0, c=-49
D = 0 - 4(1)(-49) = 196 → √196 = 14
x = [0 ± 14]/2 → x = 7 or x = -7

✔ Roots: x = 7, x = -7

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4. 5k² + 20k + 25 = k²

Bring all terms to left: 5k² - k² + 20k + 25 = 0 → 4k² + 20k + 25 = 0
a=4, b=20, c=25
D = 400 - 400 = 0 → one real root
x = [-20 ± 0]/8 = -20/8 = -2.5

✔ Root: k = -2.5

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5. x² – x – 6 = -6 – 7x

Move all to left: x² – x – 6 + 6 + 7x = 0 → x² + 6x = 0
Factor: x(x + 6) = 0 → x=0 or x=-6
Or formula: a=1, b=6, c=0
D = 36 - 0 = 36 → √36=6
x = [-6 ± 6]/2 → x=0 or x=-6

✔ Roots: x = 0, x = -6

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6. 10x – 1/x = 3

Multiply both sides by x (x ≠ 0):
10x² – 1 = 3x → 10x² – 3x – 1 = 0
a=10, b=-3, c=-1
D = 9 + 40 = 49 → √49=7
x = [3 ± 7]/20 → x=10/20=0.5 or x=-4/20=-0.2

✔ Roots: x = 0.5, x = -0.2

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7. (x+1)/(x–1) – (x–1)/(x+1) = 5/6, x≠1,-1

Find common denominator: (x–1)(x+1) = x²–1
Numerator: (x+1)² – (x–1)² = [x²+2x+1] – [x²–2x+1] = 4x
So: 4x / (x²–1) = 5/6
Cross multiply: 24x = 5(x² – 1) → 24x = 5x² – 5
→ 5x² – 24x – 5 = 0
a=5, b=-24, c=-5
D = 576 + 100 = 676 → √676=26
x = [24 ± 26]/10 → x=50/10=5 or x=-2/10=-0.2

Check restrictions: x≠1,-1 → both OK

✔ Roots: x = 5, x = -0.2

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8. x² – 2x + 5 = 0

a=1, b=-2, c=5
D = 4 - 20 = -16 → no real roots

✔ Roots: Complex

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9. √2 x² – 3x – 2√2 = 0

a=√2, b=-3, c=-2√2
D = 9 - 4(√2)(-2√2) = 9 + 8*2 = 9 + 16 = 25 → √25=5
x = [3 ± 5]/(2√2)
→ x = 8/(2√2) = 4/√2 = 2√2
→ x = -2/(2√2) = -1/√2 = -√2/2

✔ Roots: x = 2√2, x = -√2/2

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10. x² + 4ix – 4 = 0

This has imaginary coefficient — but we can still use formula.
a=1, b=4i, c=-4
D = (4i)² - 4(1)(-4) = -16 + 16 = 0 → one root
x = [-4i ± 0]/2 = -2i

✔ Root: x = -2i

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11. 3x² – 17x + 25 = 0

a=3, b=-17, c=25
D = 289 - 300 = -11 → no real roots

✔ Roots: Complex

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Final Answer:
1. x = 4 (repeated)
2. No real roots
3. x = 7, x = -7
4. k = -2.5
5. x = 0, x = -6
6. x = 0.5, x = -0.2
7. x = 5, x = -0.2
8. No real roots
9. x = 2√2, x = -√2/2
10. x = -2i
11. No real roots