Algebra Examples

$\frac{1}{2} \cdot {(x - 4)}^{2} = 8$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the left side.

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Step 1.1.1

Combine $\frac{1}{2}$ and ${(x - 4)}^{2}$ .

$\frac{{(x - 4)}^{2}}{2} = 8$

Step 1.2

Subtract $8$ from both sides of the equation.

$\frac{{(x - 4)}^{2}}{2} - 8 = 0$

Step 2

Multiply through by the least common denominator $2$ , then simplify.

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Step 2.1

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

$2 (\frac{{(x - 4)}^{2}}{2} - 8 \cdot \frac{2}{2}) = 0$

Step 2.2

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

$2 (\frac{{(x - 4)}^{2}}{2} + \frac{- 8 \cdot 2}{2}) = 0$

Step 2.3

Combine the numerators over the common denominator.

$2 (\frac{{(x - 4)}^{2} - 8 \cdot 2}{2}) = 0$

Step 2.4

Cancel the common factor of $2$ .

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Step 2.4.1

Cancel the common factor.

$2 (\frac{{(x - 4)}^{2} - 8 \cdot 2}{2}) = 0$

Step 2.4.2

Rewrite the expression.

${(x - 4)}^{2} - 8 \cdot 2 = 0$

Step 2.5

Simplify each term.

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Step 2.5.1

Rewrite ${(x - 4)}^{2}$ as $(x - 4) (x - 4)$ .

$(x - 4) (x - 4) - 8 \cdot 2 = 0$

Step 2.5.2

Expand $(x - 4) (x - 4)$ using the FOIL Method.

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Step 2.5.2.1

Apply the distributive property.

$x (x - 4) - 4 (x - 4) - 8 \cdot 2 = 0$

Step 2.5.2.2

Apply the distributive property.

$x \cdot x + x \cdot - 4 - 4 (x - 4) - 8 \cdot 2 = 0$

Step 2.5.2.3

Apply the distributive property.

$x \cdot x + x \cdot - 4 - 4 x - 4 \cdot - 4 - 8 \cdot 2 = 0$

Step 2.5.3

Simplify and combine like terms.

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Step 2.5.3.1

Simplify each term.

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Step 2.5.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 4 - 4 x - 4 \cdot - 4 - 8 \cdot 2 = 0$

Step 2.5.3.1.2

Move $- 4$ to the left of $x$ .

$x^{2} - 4 \cdot x - 4 x - 4 \cdot - 4 - 8 \cdot 2 = 0$

Step 2.5.3.1.3

Multiply $- 4$ by $- 4$ .

$x^{2} - 4 x - 4 x + 16 - 8 \cdot 2 = 0$

Step 2.5.3.2

Subtract $4 x$ from $- 4 x$ .

$x^{2} - 8 x + 16 - 8 \cdot 2 = 0$

Step 2.5.4

Multiply $- 8$ by $2$ .

$x^{2} - 8 x + 16 - 16 = 0$

Step 2.6

Combine the opposite terms in $x^{2} - 8 x + 16 - 16$ .

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Step 2.6.1

Subtract $16$ from $16$ .

$x^{2} - 8 x + 0 = 0$

Step 2.6.2

Add $x^{2} - 8 x$ and $0$ .

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

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 $a = 1$ , $b = - 8$ , and $c = 0$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (1 \cdot 0)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 1 \cdot 0}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 0$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 0}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $0$ .

$x = \frac{8 \pm \sqrt{64 + 0}}{2 \cdot 1}$

Step 5.1.3

Add $64$ and $0$ .

$x = \frac{8 \pm \sqrt{64}}{2 \cdot 1}$

Step 5.1.4

Rewrite $64$ as $8^{2}$ .

$x = \frac{8 \pm \sqrt{8^{2}}}{2 \cdot 1}$

Step 5.1.5

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

$x = \frac{8 \pm 8}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{8 \pm 8}{2}$

Step 5.3

Simplify $\frac{8 \pm 8}{2}$ .

$x = 4 \pm 4$

Step 6

The final answer is the combination of both solutions.

$x = 8, 0$