Solving Quadratic Equations Using The Quadratic Formula Worksheet - Free Printable

Explanation:
We are solving quadratic equations using the quadratic formula. The quadratic formula is:

$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$

This formula works for any quadratic equation in the form $ ax^2 + bx + c = 0 $, where $ a $, $ b $, and $ c $ are numbers, and $ a \neq 0 $.

To use the formula, we:
1. Write the equation in standard form: $ ax^2 + bx + c = 0 $
2. Identify $ a $, $ b $, and $ c $
3. Plug them into the formula
4. Simplify the expression

Let’s solve each equation step by step.

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1) $ 3n^2 - 5n - 8 = 0 $

- $ a = 3 $, $ b = -5 $, $ c = -8 $
- Discriminant: $ b^2 - 4ac = (-5)^2 - 4(3)(-8) = 25 + 96 = 121 $
- $ \sqrt{121} = 11 $
- $ n = \frac{-(-5) \pm 11}{2(3)} = \frac{5 \pm 11}{6} $
- $ n = \frac{5 + 11}{6} = \frac{16}{6} = \frac{8}{3} $
- $ n = \frac{5 - 11}{6} = \frac{-6}{6} = -1 $

Answer: $ n = -1 $ or $ n = \frac{8}{3} $

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2) $ x^2 + 10x + 21 = 0 $

- $ a = 1 $, $ b = 10 $, $ c = 21 $
- Discriminant: $ 10^2 - 4(1)(21) = 100 - 84 = 16 $
- $ \sqrt{16} = 4 $
- $ x = \frac{-10 \pm 4}{2(1)} = \frac{-10 \pm 4}{2} $
- $ x = \frac{-10 + 4}{2} = \frac{-6}{2} = -3 $
- $ x = \frac{-10 - 4}{2} = \frac{-14}{2} = -7 $

Answer: $ x = -7 $ or $ x = -3 $

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3) $ 10x^2 - 9x + 6 = 0 $

- $ a = 10 $, $ b = -9 $, $ c = 6 $
- Discriminant: $ (-9)^2 - 4(10)(6) = 81 - 240 = -159 $
- Negative discriminant → no real solutions

Answer: No real solutions

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4) $ p^2 - 9 = 0 $

- $ a = 1 $, $ b = 0 $, $ c = -9 $
- Discriminant: $ 0^2 - 4(1)(-9) = 0 + 36 = 36 $
- $ \sqrt{36} = 6 $
- $ p = \frac{-0 \pm 6}{2(1)} = \frac{\pm 6}{2} $
- $ p = 3 $ or $ p = -3 $

Answer: $ p = -3 $ or $ p = 3 $

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5) $ 6x^2 - 12x + 1 = 0 $

- $ a = 6 $, $ b = -12 $, $ c = 1 $
- Discriminant: $ (-12)^2 - 4(6)(1) = 144 - 24 = 120 $
- $ \sqrt{120} = \sqrt{4 \cdot 30} = 2\sqrt{30} $
- $ x = \frac{12 \pm 2\sqrt{30}}{12} = \frac{6 \pm \sqrt{30}}{6} $

Answer: $ x = \frac{6 \pm \sqrt{30}}{6} $

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6) $ 6n^2 - 11 = 0 $

- $ a = 6 $, $ b = 0 $, $ c = -11 $
- Discriminant: $ 0^2 - 4(6)(-11) = 0 + 264 = 264 $
- $ \sqrt{264} = \sqrt{4 \cdot 66} = 2\sqrt{66} $
- $ n = \frac{0 \pm 2\sqrt{66}}{12} = \frac{\pm 2\sqrt{66}}{12} = \frac{\pm \sqrt{66}}{6} $

Answer: $ n = \frac{-\sqrt{66}}{6} $ or $ n = \frac{\sqrt{66}}{6} $

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7) $ 2n^2 + 5n - 9 = 0 $

- $ a = 2 $, $ b = 5 $, $ c = -9 $
- Discriminant: $ 5^2 - 4(2)(-9) = 25 + 72 = 97 $
- $ \sqrt{97} $ is not a perfect square
- $ n = \frac{-5 \pm \sqrt{97}}{4} $

Answer: $ n = \frac{-5 \pm \sqrt{97}}{4} $

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8) $ 3x^2 - 6x - 23 = 0 $

- $ a = 3 $, $ b = -6 $, $ c = -23 $
- Discriminant: $ (-6)^2 - 4(3)(-23) = 36 + 276 = 312 $
- $ \sqrt{312} = \sqrt{4 \cdot 78} = 2\sqrt{78} $
- $ x = \frac{6 \pm 2\sqrt{78}}{6} = \frac{3 \pm \sqrt{78}}{3} $

Answer: $ x = \frac{3 \pm \sqrt{78}}{3} $

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9) $ 6k^2 + 12k - 15 = -10 $

- First, move all terms to one side: $ 6k^2 + 12k - 15 + 10 = 0 $
- $ 6k^2 + 12k - 5 = 0 $
- $ a = 6 $, $ b = 12 $, $ c = -5 $
- Discriminant: $ 12^2 - 4(6)(-5) = 144 + 120 = 264 $
- $ \sqrt{264} = 2\sqrt{66} $
- $ k = \frac{-12 \pm 2\sqrt{66}}{12} = \frac{-6 \pm \sqrt{66}}{6} $

Answer: $ k = \frac{-6 \pm \sqrt{66}}{6} $

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10) $ 8x^2 - 14 = -11 $

- Move terms: $ 8x^2 - 14 + 11 = 0 $
- $ 8x^2 - 3 = 0 $
- $ a = 8 $, $ b = 0 $, $ c = -3 $
- Discriminant: $ 0^2 - 4(8)(-3) = 96 $
- $ \sqrt{96} = \sqrt{16 \cdot 6} = 4\sqrt{6} $
- $ x = \frac{0 \pm 4\sqrt{6}}{16} = \frac{\pm 4\sqrt{6}}{16} = \frac{\pm \sqrt{6}}{4} $

Answer: $ x = \frac{-\sqrt{6}}{4} $ or $ x = \frac{\sqrt{6}}{4} $

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11) $ 6k^2 + 2k + 9 = -3 $

- Move terms: $ 6k^2 + 2k + 9 + 3 = 0 $
- $ 6k^2 + 2k + 12 = 0 $
- Divide all terms by 2: $ 3k^2 + k + 6 = 0 $
- $ a = 3 $, $ b = 1 $, $ c = 6 $
- Discriminant: $ 1^2 - 4(3)(6) = 1 - 72 = -71 $
- Negative discriminant → no real solutions

Answer: No real solutions

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12) $ 12p^2 + 9p - 30 = -10 $

- Move terms: $ 12p^2 + 9p - 30 + 10 = 0 $
- $ 12p^2 + 9p - 20 = 0 $
- $ a = 12 $, $ b = 9 $, $ c = -20 $
- Discriminant: $ 9^2 - 4(12)(-20) = 81 + 960 = 1041 $
- $ \sqrt{1041} $ is not a perfect square
- $ p = \frac{-9 \pm \sqrt{1041}}{24} $

Answer: $ p = \frac{-9 \pm \sqrt{1041}}{24} $

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13) $ 3x^2 = -7x + 136 $

- Move all terms to one side: $ 3x^2 + 7x - 136 = 0 $
- $ a = 3 $, $ b = 7 $, $ c = -136 $
- Discriminant: $ 7^2 - 4(3)(-136) = 49 + 1632 = 1681 $
- $ \sqrt{1681} = 41 $
- $ x = \frac{-7 \pm 41}{6} $
- $ x = \frac{-7 + 41}{6} = \frac{34}{6} = \frac{17}{3} $
- $ x = \frac{-7 - 41}{6} = \frac{-48}{6} = -8 $

Answer: $ x = -8 $ or $ x = \frac{17}{3} $

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14) $ 3n^2 = -n + 14 $

- Move terms: $ 3n^2 + n - 14 = 0 $
- $ a = 3 $, $ b = 1 $, $ c = -14 $
- Discriminant: $ 1^2 - 4(3)(-14) = 1 + 168 = 169 $
- $ \sqrt{169} = 13 $
- $ n = \frac{-1 \pm 13}{6} $
- $ n = \frac{-1 + 13}{6} = \frac{12}{6} = 2 $
- $ n = \frac{-1 - 13}{6} = \frac{-14}{6} = -\frac{7}{3} $

Answer: $ n = 2 $ or $ n = -\frac{7}{3} $

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15) $ 6v^2 + 3 = -2v $

- Move terms: $ 6v^2 + 2v + 3 = 0 $
- $ a = 6 $, $ b = 2 $, $ c = 3 $
- Discriminant: $ 2^2 - 4(6)(3) = 4 - 72 = -68 $
- Negative discriminant → no real solutions

Answer: No real solutions

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16) $ 9p^2 - 7 = 9p $

- Move terms: $ 9p^2 - 9p - 7 = 0 $
- $ a = 9 $, $ b = -9 $, $ c = -7 $
- Discriminant: $ (-9)^2 - 4(9)(-7) = 81 + 252 = 333 $
- $ \sqrt{333} = \sqrt{9 \cdot 37} = 3\sqrt{37} $
- $ p = \frac{9 \pm 3\sqrt{37}}{18} = \frac{3 \pm \sqrt{37}}{6} $

Answer: $ p = \frac{3 \pm \sqrt{37}}{6} $

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17) $ 11k^2 + 4k - 52 = 10k^2 - 7 $

- Move all terms to one side: $ 11k^2 - 10k^2 + 4k - 52 + 7 = 0 $
- $ k^2 + 4k - 45 = 0 $
- $ a = 1 $, $ b = 4 $, $ c = -45 $
- Discriminant: $ 4^2 - 4(1)(-45) = 16 + 180 = 196 $
- $ \sqrt{196} = 14 $
- $ k = \frac{-4 \pm 14}{2} $
- $ k = \frac{-4 + 14}{2} = \frac{10}{2} = 5 $
- $ k = \frac{-4 - 14}{2} = \frac{-18}{2} = -9 $

Answer: $ k = -9 $ or $ k = 5 $

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18) $ -4a^2 + 18a - 15 = -7a^2 + 9a $

- Move all terms to one side: $ -4a^2 + 7a^2 + 18a - 9a - 15 = 0 $
- $ 3a^2 + 9a - 15 = 0 $
- Divide all terms by 3: $ a^2 + 3a - 5 = 0 $
- $ a = 1 $, $ b = 3 $, $ c = -5 $
- Discriminant: $ 3^2 - 4(1)(-5) = 9 + 20 = 29 $
- $ \sqrt{29} $ is not a perfect square
- $ a = \frac{-3 \pm \sqrt{29}}{2} $

Answer: $ a = \frac{-3 \pm \sqrt{29}}{2} $

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19) $ -4n(n - 2) = 6(n + 3) - 11n^2 $

- Expand both sides:
- Left: $ -4n^2 + 8n $
- Right: $ 6n + 18 - 11n^2 $
- Equation: $ -4n^2 + 8n = 6n + 18 - 11n^2 $
- Move all terms to one side: $ -4n^2 + 11n^2 + 8n - 6n - 18 = 0 $
- $ 7n^2 + 2n - 18 = 0 $
- $ a = 7 $, $ b = 2 $, $ c = -18 $
- Discriminant: $ 2^2 - 4(7)(-18) = 4 + 504 = 508 $
- $ \sqrt{508} = \sqrt{4 \cdot 127} = 2\sqrt{127} $
- $ n = \frac{-2 \pm 2\sqrt{127}}{14} = \frac{-1 \pm \sqrt{127}}{7} $

Answer: $ n = \frac{-1 \pm \sqrt{127}}{7} $

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20) $ x(x - 3) = -7 - 10x $

- Expand: $ x^2 - 3x = -7 - 10x $
- Move all terms: $ x^2 - 3x + 10x + 7 = 0 $
- $ x^2 + 7x + 7 = 0 $
- $ a = 1 $, $ b = 7 $, $ c = 7 $
- Discriminant: $ 7^2 - 4(1)(7) = 49 - 28 = 21 $
- $ \sqrt{21} $ is not a perfect square
- $ x = \frac{-7 \pm \sqrt{21}}{2} $

Answer: $ x = \frac{-7 \pm \sqrt{21}}{2} $

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Final Answer:
1) $ n = -1 $ or $ n = \frac{8}{3} $
2) $ x = -7 $ or $ x = -3 $
3) No real solutions
4) $ p = -3 $ or $ p = 3 $
5) $ x = \frac{6 \pm \sqrt{30}}{6} $
6) $ n = \frac{-\sqrt{66}}{6} $ or $ n = \frac{\sqrt{66}}{6} $
7) $ n = \frac{-5 \pm \sqrt{97}}{4} $
8) $ x = \frac{3 \pm \sqrt{78}}{3} $
9) $ k = \frac{-6 \pm \sqrt{66}}{6} $
10) $ x = \frac{-\sqrt{6}}{4} $ or $ x = \frac{\sqrt{6}}{4} $
11) No real solutions
12) $ p = \frac{-9 \pm \sqrt{1041}}{24} $
13) $ x = -8 $ or $ x = \frac{17}{3} $
14) $ n = 2 $ or $ n = -\frac{7}{3} $
15) No real solutions
16) $ p = \frac{3 \pm \sqrt{37}}{6} $
17) $ k = -9 $ or $ k = 5 $
18) $ a = \frac{-3 \pm \sqrt{29}}{2} $
19) $ n = \frac{-1 \pm \sqrt{127}}{7} $
20) $ x = \frac{-7 \pm \sqrt{21}}{2} $