Quadratic Formula Worksheets - Math Monks - Free Printable

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

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

We will go through each problem step by step.

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1. $ 4x^2 - 12x + 9 = 0 $

Identify coefficients:
- $ a = 4 $
- $ b = -12 $
- $ c = 9 $

Plug into the formula:

$$
x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(4)(9)}}{2(4)}
= \frac{12 \pm \sqrt{144 - 144}}{8}
= \frac{12 \pm \sqrt{0}}{8}
= \frac{12}{8} = \frac{3}{2}
$$

✔ Solution: $ x = \frac{3}{2} $ (double root)

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2. $ 6x^2 + 5x - 25 = 0 $

Coefficients:
- $ a = 6 $
- $ b = 5 $
- $ c = -25 $

$$
x = \frac{-5 \pm \sqrt{(5)^2 - 4(6)(-25)}}{2(6)}
= \frac{-5 \pm \sqrt{25 + 600}}{12}
= \frac{-5 \pm \sqrt{625}}{12}
= \frac{-5 \pm 25}{12}
$$

Now calculate both solutions:

- $ x = \frac{-5 + 25}{12} = \frac{20}{12} = \frac{5}{3} $
- $ x = \frac{-5 - 25}{12} = \frac{-30}{12} = -\frac{5}{2} $

✔ Solutions: $ x = \frac{5}{3},\; x = -\frac{5}{2} $

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3. $ 2x^2 + 39 + 18x = 0 $

First, rewrite in standard form:
$ 2x^2 + 18x + 39 = 0 $

Coefficients:
- $ a = 2 $
- $ b = 18 $
- $ c = 39 $

$$
x = \frac{-18 \pm \sqrt{(18)^2 - 4(2)(39)}}{2(2)}
= \frac{-18 \pm \sqrt{324 - 312}}{4}
= \frac{-18 \pm \sqrt{12}}{4}
$$

Simplify $ \sqrt{12} = 2\sqrt{3} $:

$$
x = \frac{-18 \pm 2\sqrt{3}}{4} = \frac{-9 \pm \sqrt{3}}{2}
$$

✔ Solutions: $ x = \frac{-9 + \sqrt{3}}{2},\; x = \frac{-9 - \sqrt{3}}{2} $

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4. $ 2x^2 + 15x + 28 = 0 $

Coefficients:
- $ a = 2 $
- $ b = 15 $
- $ c = 28 $

$$
x = \frac{-15 \pm \sqrt{(15)^2 - 4(2)(28)}}{2(2)}
= \frac{-15 \pm \sqrt{225 - 224}}{4}
= \frac{-15 \pm \sqrt{1}}{4}
= \frac{-15 \pm 1}{4}
$$

So:

- $ x = \frac{-15 + 1}{4} = \frac{-14}{4} = -\frac{7}{2} $
- $ x = \frac{-15 - 1}{4} = \frac{-16}{4} = -4 $

✔ Solutions: $ x = -\frac{7}{2},\; x = -4 $

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5. $ 9x^2 + 4x - 16 = 0 $

Coefficients:
- $ a = 9 $
- $ b = 4 $
- $ c = -16 $

$$
x = \frac{-4 \pm \sqrt{(4)^2 - 4(9)(-16)}}{2(9)}
= \frac{-4 \pm \sqrt{16 + 576}}{18}
= \frac{-4 \pm \sqrt{592}}{18}
$$

Simplify $ \sqrt{592} $:

$ 592 = 16 \times 37 $, so $ \sqrt{592} = 4\sqrt{37} $

$$
x = \frac{-4 \pm 4\sqrt{37}}{18} = \frac{-2 \pm 2\sqrt{37}}{9}
$$

✔ Solutions: $ x = \frac{-2 + 2\sqrt{37}}{9},\; x = \frac{-2 - 2\sqrt{37}}{9} $

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6. $ 15x^2 + 4x - 4 = 0 $

Coefficients:
- $ a = 15 $
- $ b = 4 $
- $ c = -4 $

$$
x = \frac{-4 \pm \sqrt{(4)^2 - 4(15)(-4)}}{2(15)}
= \frac{-4 \pm \sqrt{16 + 240}}{30}
= \frac{-4 \pm \sqrt{256}}{30}
= \frac{-4 \pm 16}{30}
$$

Now compute:

- $ x = \frac{-4 + 16}{30} = \frac{12}{30} = \frac{2}{5} $
- $ x = \frac{-4 - 16}{30} = \frac{-20}{30} = -\frac{2}{3} $

✔ Solutions: $ x = \frac{2}{5},\; x = -\frac{2}{3} $

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✔ Final Answers:

1. $ x = \frac{3}{2} $
2. $ x = \frac{5}{3},\; x = -\frac{5}{2} $
3. $ x = \frac{-9 \pm \sqrt{3}}{2} $
4. $ x = -\frac{7}{2},\; x = -4 $
5. $ x = \frac{-2 \pm 2\sqrt{37}}{9} $
6. $ x = \frac{2}{5},\; x = -\frac{2}{3} $

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