Algebra Examples

$g (x) = 13 - x^{2}$

Step 1

Add $x^{2}$ to both sides of the equation.

$g x + x^{2} = 13$

Step 2

Subtract $13$ from both sides of the equation.

$g x + x^{2} - 13 = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = 1$ , $b = g$ , and $c = - 13$ into the quadratic formula and solve for $x$ .

$\frac{- g \pm \sqrt{g^{2} - 4 \cdot (1 \cdot - 13)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Multiply $- 4 \cdot 1 \cdot - 13$ .

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Step 5.1.1

Multiply $- 4$ by $1$ .

$x = \frac{- g \pm \sqrt{g^{2} - 4 \cdot - 13}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4$ by $- 13$ .

$x = \frac{- g \pm \sqrt{g^{2} + 52}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{- g \pm \sqrt{g^{2} + 52}}{2}$

Step 6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.1

Multiply $- 4 \cdot 1 \cdot - 13$ .

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Step 6.1.1

Multiply $- 4$ by $1$ .

$x = \frac{- g \pm \sqrt{g^{2} - 4 \cdot - 13}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4$ by $- 13$ .

$x = \frac{- g \pm \sqrt{g^{2} + 52}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{- g \pm \sqrt{g^{2} + 52}}{2}$

Step 6.3

Change the $\pm$ to $+$ .

$x = \frac{- g + \sqrt{g^{2} + 52}}{2}$

Step 6.4

Factor $- 1$ out of $- g$ .

$x = \frac{- (g) + \sqrt{g^{2} + 52}}{2}$

Step 6.5

Factor $- 1$ out of $\sqrt{g^{2} + 52}$ .

$x = \frac{- (g) - 1 (- \sqrt{g^{2} + 52})}{2}$

Step 6.6

Factor $- 1$ out of $- (g) - 1 (- \sqrt{g^{2} + 52})$ .

$x = \frac{- (g - \sqrt{g^{2} + 52})}{2}$

Step 6.7

Rewrite $- (g - \sqrt{g^{2} + 52})$ as $- 1 (g - \sqrt{g^{2} + 52})$ .

$x = \frac{- 1 (g - \sqrt{g^{2} + 52})}{2}$

Step 6.8

Move the negative in front of the fraction.

$x = - \frac{g - \sqrt{g^{2} + 52}}{2}$

Step 7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 7.1

Multiply $- 4 \cdot 1 \cdot - 13$ .

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Step 7.1.1

Multiply $- 4$ by $1$ .

$x = \frac{- g \pm \sqrt{g^{2} - 4 \cdot - 13}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4$ by $- 13$ .

$x = \frac{- g \pm \sqrt{g^{2} + 52}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$x = \frac{- g \pm \sqrt{g^{2} + 52}}{2}$

Step 7.3

Change the $\pm$ to $-$ .

$x = \frac{- g - \sqrt{g^{2} + 52}}{2}$

Step 7.4

Factor $- 1$ out of $- g$ .

$x = \frac{- (g) - \sqrt{g^{2} + 52}}{2}$

Step 7.5

Factor $- 1$ out of $- \sqrt{g^{2} + 52}$ .

$x = \frac{- (g) - (\sqrt{g^{2} + 52})}{2}$

Step 7.6

Factor $- 1$ out of $- (g) - (\sqrt{g^{2} + 52})$ .

$x = \frac{- (g + \sqrt{g^{2} + 52})}{2}$

Step 7.7

Rewrite $- (g + \sqrt{g^{2} + 52})$ as $- 1 (g + \sqrt{g^{2} + 52})$ .

$x = \frac{- 1 (g + \sqrt{g^{2} + 52})}{2}$

Step 7.8

Move the negative in front of the fraction.

$x = - \frac{g + \sqrt{g^{2} + 52}}{2}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{g - \sqrt{g^{2} + 52}}{2}$

$x = - \frac{g + \sqrt{g^{2} + 52}}{2}$