How To Solve Quadratic Equations By Factoring - Quick & Simple! | Algebra Online Course - Free Printable

Let's solve each of the quadratic equations shown in the image step by step. We'll use factoring, the quadratic formula, or completing the square as appropriate.

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1. $ x^2 - 2x - 15 = 0 $

We look for two numbers that multiply to $-15$ and add to $-2$.
These numbers are: $-5$ and $3$.

$$
x^2 - 2x - 15 = (x - 5)(x + 3) = 0
$$

Set each factor equal to zero:

$$
x - 5 = 0 \Rightarrow x = 5 \\
x + 3 = 0 \Rightarrow x = -3
$$

✔ Solutions: $ x = 5, -3 $

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2. $ x^2 - 49 = 0 $

This is a difference of squares:

$$
x^2 - 49 = (x - 7)(x + 7) = 0
$$

$$
x - 7 = 0 \Rightarrow x = 7 \\
x + 7 = 0 \Rightarrow x = -7
$$

✔ Solutions: $ x = 7, -7 $

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3. $ x^2 + 3x - 28 = 0 $

Find two numbers that multiply to $-28$ and add to $3$:
These are $7$ and $-4$

$$
x^2 + 3x - 28 = (x + 7)(x - 4) = 0
$$

$$
x + 7 = 0 \Rightarrow x = -7 \\
x - 4 = 0 \Rightarrow x = 4
$$

✔ Solutions: $ x = -7, 4 $

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4. $ 3x^2 - 75 = 0 $

First, divide both sides by 3:

$$
x^2 - 25 = 0
$$

This is again a difference of squares:

$$
x^2 - 25 = (x - 5)(x + 5) = 0
$$

$$
x = 5, -5
$$

✔ Solutions: $ x = 5, -5 $

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5. $ 8x^2 + 2x - 15 = 0 $

Use the quadratic formula:

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

Here, $ a = 8 $, $ b = 2 $, $ c = -15 $

$$
x = \frac{-2 \pm \sqrt{(2)^2 - 4(8)(-15)}}{2(8)} = \frac{-2 \pm \sqrt{4 + 480}}{16} = \frac{-2 \pm \sqrt{484}}{16}
$$

$$
\sqrt{484} = 22
$$

$$
x = \frac{-2 \pm 22}{16}
$$

Now compute both solutions:

- $ x = \frac{-2 + 22}{16} = \frac{20}{16} = \frac{5}{4} $
- $ x = \frac{-2 - 22}{16} = \frac{-24}{16} = -\frac{3}{2} $

✔ Solutions: $ x = \frac{5}{4}, -\frac{3}{2} $

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6. $ 9x^2 - 64 = 0 $

This is a difference of squares:

$$
9x^2 - 64 = (3x)^2 - (8)^2 = (3x - 8)(3x + 8) = 0
$$

Set each factor to zero:

$$
3x - 8 = 0 \Rightarrow x = \frac{8}{3} \\
3x + 8 = 0 \Rightarrow x = -\frac{8}{3}
$$

✔ Solutions: $ x = \frac{8}{3}, -\frac{8}{3} $

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✔ Final Answers Summary:

| Equation | Solutions |
|--------|-----------|
| $ x^2 - 2x - 15 = 0 $ | $ x = 5, -3 $ |
| $ x^2 - 49 = 0 $ | $ x = 7, -7 $ |
| $ x^2 + 3x - 28 = 0 $ | $ x = -7, 4 $ |
| $ 3x^2 - 75 = 0 $ | $ x = 5, -5 $ |
| $ 8x^2 + 2x - 15 = 0 $ | $ x = \frac{5}{4}, -\frac{3}{2} $ |
| $ 9x^2 - 64 = 0 $ | $ x = \frac{8}{3}, -\frac{8}{3} $ |

Let me know if you'd like these explained visually or with graphs!