Algebra Examples

$\frac{2}{3} \cdot {(x + 8)}^{2} - 66 = 0$

Step 1

Simplify the left side.

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Step 1.1

Combine $\frac{2}{3}$ and ${(x + 8)}^{2}$ .

$\frac{2 {(x + 8)}^{2}}{3} - 66 = 0$

Step 2

Multiply through by the least common denominator $3$ , then simplify.

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Step 2.1

To write $- 66$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$3 (\frac{2 {(x + 8)}^{2}}{3} - 66 \cdot \frac{3}{3}) = 0$

Step 2.2

Combine $- 66$ and $\frac{3}{3}$ .

$3 (\frac{2 {(x + 8)}^{2}}{3} + \frac{- 66 \cdot 3}{3}) = 0$

Step 2.3

Combine the numerators over the common denominator.

$3 (\frac{2 {(x + 8)}^{2} - 66 \cdot 3}{3}) = 0$

Step 2.4

Cancel the common factor of $3$ .

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Step 2.4.1

Cancel the common factor.

$3 (\frac{2 {(x + 8)}^{2} - 66 \cdot 3}{3}) = 0$

Step 2.4.2

Rewrite the expression.

$2 {(x + 8)}^{2} - 66 \cdot 3 = 0$

Step 2.5

Simplify each term.

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Step 2.5.1

Rewrite ${(x + 8)}^{2}$ as $(x + 8) (x + 8)$ .

$2 ((x + 8) (x + 8)) - 66 \cdot 3 = 0$

Step 2.5.2

Expand $(x + 8) (x + 8)$ using the FOIL Method.

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Step 2.5.2.1

Apply the distributive property.

$2 (x (x + 8) + 8 (x + 8)) - 66 \cdot 3 = 0$

Step 2.5.2.2

Apply the distributive property.

$2 (x \cdot x + x \cdot 8 + 8 (x + 8)) - 66 \cdot 3 = 0$

Step 2.5.2.3

Apply the distributive property.

$2 (x \cdot x + x \cdot 8 + 8 x + 8 \cdot 8) - 66 \cdot 3 = 0$

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 $x$ .

$2 (x^{2} + x \cdot 8 + 8 x + 8 \cdot 8) - 66 \cdot 3 = 0$

Step 2.5.3.1.2

Move $8$ to the left of $x$ .

$2 (x^{2} + 8 \cdot x + 8 x + 8 \cdot 8) - 66 \cdot 3 = 0$

Step 2.5.3.1.3

Multiply $8$ by $8$ .

$2 (x^{2} + 8 x + 8 x + 64) - 66 \cdot 3 = 0$

Step 2.5.3.2

Add $8 x$ and $8 x$ .

$2 (x^{2} + 16 x + 64) - 66 \cdot 3 = 0$

Step 2.5.4

Apply the distributive property.

$2 x^{2} + 2 (16 x) + 2 \cdot 64 - 66 \cdot 3 = 0$

Step 2.5.5

Simplify.

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Step 2.5.5.1

Multiply $16$ by $2$ .

$2 x^{2} + 32 x + 2 \cdot 64 - 66 \cdot 3 = 0$

Step 2.5.5.2

Multiply $2$ by $64$ .

$2 x^{2} + 32 x + 128 - 66 \cdot 3 = 0$

Step 2.5.6

Multiply $- 66$ by $3$ .

$2 x^{2} + 32 x + 128 - 198 = 0$

Step 2.6

Subtract $198$ from $128$ .

$2 x^{2} + 32 x - 70 = 0$

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 $a = 2$ , $b = 32$ , and $c = - 70$ into the quadratic formula and solve for $x$ .

$\frac{- 32 \pm \sqrt{32^{2} - 4 \cdot (2 \cdot - 70)}}{2 \cdot 2}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $32$ to the power of $2$ .

$x = \frac{- 32 \pm \sqrt{1024 - 4 \cdot 2 \cdot - 70}}{2 \cdot 2}$

Step 5.1.2

Multiply $- 4 \cdot 2 \cdot - 70$ .

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Step 5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 32 \pm \sqrt{1024 - 8 \cdot - 70}}{2 \cdot 2}$

Step 5.1.2.2

Multiply $- 8$ by $- 70$ .

$x = \frac{- 32 \pm \sqrt{1024 + 560}}{2 \cdot 2}$

Step 5.1.3

Add $1024$ and $560$ .

$x = \frac{- 32 \pm \sqrt{1584}}{2 \cdot 2}$

Step 5.1.4

Rewrite $1584$ as $12^{2} \cdot 11$ .

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Step 5.1.4.1

Factor $144$ out of $1584$ .

$x = \frac{- 32 \pm \sqrt{144 (11)}}{2 \cdot 2}$

Step 5.1.4.2

Rewrite $144$ as $12^{2}$ .

$x = \frac{- 32 \pm \sqrt{12^{2} \cdot 11}}{2 \cdot 2}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{- 32 \pm 12 \sqrt{11}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$x = \frac{- 32 \pm 12 \sqrt{11}}{4}$

Step 5.3

Simplify $\frac{- 32 \pm 12 \sqrt{11}}{4}$ .

$x = - 8 \pm 3 \sqrt{11}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - 8 \pm 3 \sqrt{11}$

Decimal Form:

$x = 1.94987437 \dots, - 17.94987437 \dots$