Algebra Examples

$\frac{1}{2} x^{2} - 8 = 0$

Step 1

Combine $\frac{1}{2}$ and $x^{2}$ .

$\frac{x^{2}}{2} - 8 = 0$

Step 2

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x^{2}}{2} - 8 \cdot \frac{2}{2} = 0$

Step 3

Combine $- 8$ and $\frac{2}{2}$ .

$\frac{x^{2}}{2} + \frac{- 8 \cdot 2}{2} = 0$

Step 4

Combine the numerators over the common denominator.

$\frac{x^{2} - 8 \cdot 2}{2} = 0$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $- 8$ by $2$ .

$\frac{x^{2} - 16}{2} = 0$

Step 5.2

Rewrite $x^{2} - 16$ in a factored form.

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Step 5.2.1

Rewrite $16$ as $4^{2}$ .

$\frac{x^{2} - 4^{2}}{2} = 0$

Step 5.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 4$ .

$\frac{(x + 4) (x - 4)}{2} = 0$

Step 6

Set the numerator equal to zero.

$(x + 4) (x - 4) = 0$

Step 7

Solve the equation for $x$ .

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Step 7.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 4 = 0$

$x - 4 = 0$

Step 7.2

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 7.2.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 7.2.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 7.3

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 7.3.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 7.3.2

Add $4$ to both sides of the equation.

$x = 4$

Step 7.4

The final solution is all the values that make $(x + 4) (x - 4) = 0$ true.

$x = - 4, 4$