Algebra Examples

$8 x = x^{2} + 12$

Step 1

Subtract $x^{2}$ from both sides of the equation.

$8 x - x^{2} = 12$

Step 2

Divide each term in $8 x - x^{2} = 12$ by $- 1$ and simplify.

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Step 2.1

Divide each term in $8 x - x^{2} = 12$ by $- 1$ .

$\frac{8 x}{- 1} + \frac{- x^{2}}{- 1} = \frac{12}{- 1}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Move the negative one from the denominator of $\frac{8 x}{- 1}$ .

$- 1 \cdot (8 x) + \frac{- x^{2}}{- 1} = \frac{12}{- 1}$

Step 2.2.1.2

Rewrite $- 1 \cdot (8 x)$ as $- (8 x)$ .

$- (8 x) + \frac{- x^{2}}{- 1} = \frac{12}{- 1}$

Step 2.2.1.3

Multiply $8$ by $- 1$ .

$- 8 x + \frac{- x^{2}}{- 1} = \frac{12}{- 1}$

Step 2.2.1.4

Dividing two negative values results in a positive value.

$- 8 x + \frac{x^{2}}{1} = \frac{12}{- 1}$

Step 2.2.1.5

Divide $x^{2}$ by $1$ .

$- 8 x + x^{2} = \frac{12}{- 1}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $12$ by $- 1$ .

$- 8 x + x^{2} = - 12$

Step 3

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(- 4)}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} - 8 x + {(- 4)}^{2} = - 12 + {(- 4)}^{2}$

Step 5

Simplify the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Raise $- 4$ to the power of $2$ .

$x^{2} - 8 x + 16 = - 12 + {(- 4)}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- 12 + {(- 4)}^{2}$ .

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Step 5.2.1.1

Raise $- 4$ to the power of $2$ .

$x^{2} - 8 x + 16 = - 12 + 16$

Step 5.2.1.2

Add $- 12$ and $16$ .

$x^{2} - 8 x + 16 = 4$

Step 6

Factor the perfect trinomial square into ${(x - 4)}^{2}$ .

${(x - 4)}^{2} = 4$

Step 7

Solve the equation for $x$ .

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Step 7.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x - 4 = \pm \sqrt{4}$

Step 7.2

Simplify $\pm \sqrt{4}$ .

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Step 7.2.1

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

$x - 4 = \pm \sqrt{2^{2}}$

Step 7.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x - 4 = \pm 2$

Step 7.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 7.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x - 4 = 2$

Step 7.3.2

Move all terms not containing $x$ to the right side of the equation.

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Step 7.3.2.1

Add $4$ to both sides of the equation.

$x = 2 + 4$

Step 7.3.2.2

Add $2$ and $4$ .

$x = 6$

Step 7.3.3

Next, use the negative value of the $\pm$ to find the second solution.

$x - 4 = - 2$

Step 7.3.4

Move all terms not containing $x$ to the right side of the equation.

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Step 7.3.4.1

Add $4$ to both sides of the equation.

$x = - 2 + 4$

Step 7.3.4.2

Add $- 2$ and $4$ .

$x = 2$

Step 7.3.5

The complete solution is the result of both the positive and negative portions of the solution.

$x = 6, 2$