Precalculus Examples

$\frac{7 x^{2}}{8} = 2 (x - 1)$

Step 1

Multiply both sides by $8$ .

$\frac{7 x^{2}}{8} \cdot 8 = 2 (x - 1) \cdot 8$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $8$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{7 x^{2}}{8} \cdot 8 = 2 (x - 1) \cdot 8$

Step 2.1.1.2

Rewrite the expression.

$7 x^{2} = 2 (x - 1) \cdot 8$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $2 (x - 1) \cdot 8$ .

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Step 2.2.1.1

Apply the distributive property.

$7 x^{2} = (2 x + 2 \cdot - 1) \cdot 8$

Step 2.2.1.2

Multiply $2$ by $- 1$ .

$7 x^{2} = (2 x - 2) \cdot 8$

Step 2.2.1.3

Apply the distributive property.

$7 x^{2} = 2 x \cdot 8 - 2 \cdot 8$

Step 2.2.1.4

Multiply.

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Step 2.2.1.4.1

Multiply $8$ by $2$ .

$7 x^{2} = 16 x - 2 \cdot 8$

Step 2.2.1.4.2

Multiply $- 2$ by $8$ .

$7 x^{2} = 16 x - 16$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $16 x$ from both sides of the equation.

$7 x^{2} - 16 x = - 16$

Step 3.2

Add $16$ to both sides of the equation.

$7 x^{2} - 16 x + 16 = 0$

Step 3.3

Use the quadratic formula to find the solutions.

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

Step 3.4

Substitute the values $a = 7$ , $b = - 16$ , and $c = 16$ into the quadratic formula and solve for $x$ .

$\frac{16 \pm \sqrt{{(- 16)}^{2} - 4 \cdot (7 \cdot 16)}}{2 \cdot 7}$

Step 3.5

Simplify.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Raise $- 16$ to the power of $2$ .

$x = \frac{16 \pm \sqrt{256 - 4 \cdot 7 \cdot 16}}{2 \cdot 7}$

Step 3.5.1.2

Multiply $- 4 \cdot 7 \cdot 16$ .

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Step 3.5.1.2.1

Multiply $- 4$ by $7$ .

$x = \frac{16 \pm \sqrt{256 - 28 \cdot 16}}{2 \cdot 7}$

Step 3.5.1.2.2

Multiply $- 28$ by $16$ .

$x = \frac{16 \pm \sqrt{256 - 448}}{2 \cdot 7}$

Step 3.5.1.3

Subtract $448$ from $256$ .

$x = \frac{16 \pm \sqrt{- 192}}{2 \cdot 7}$

Step 3.5.1.4

Rewrite $- 192$ as $- 1 (192)$ .

$x = \frac{16 \pm \sqrt{- 1 \cdot 192}}{2 \cdot 7}$

Step 3.5.1.5

Rewrite $\sqrt{- 1 (192)}$ as $\sqrt{- 1} \cdot \sqrt{192}$ .

$x = \frac{16 \pm \sqrt{- 1} \cdot \sqrt{192}}{2 \cdot 7}$

Step 3.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{16 \pm i \cdot \sqrt{192}}{2 \cdot 7}$

Step 3.5.1.7

Rewrite $192$ as $8^{2} \cdot 3$ .

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Step 3.5.1.7.1

Factor $64$ out of $192$ .

$x = \frac{16 \pm i \cdot \sqrt{64 (3)}}{2 \cdot 7}$

Step 3.5.1.7.2

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

$x = \frac{16 \pm i \cdot \sqrt{8^{2} \cdot 3}}{2 \cdot 7}$

Step 3.5.1.8

Pull terms out from under the radical.

$x = \frac{16 \pm i \cdot (8 \sqrt{3})}{2 \cdot 7}$

Step 3.5.1.9

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

$x = \frac{16 \pm 8 i \sqrt{3}}{2 \cdot 7}$

Step 3.5.2

Multiply $2$ by $7$ .

$x = \frac{16 \pm 8 i \sqrt{3}}{14}$

Step 3.5.3

Simplify $\frac{16 \pm 8 i \sqrt{3}}{14}$ .

$x = \frac{8 \pm 4 i \sqrt{3}}{7}$

Step 3.6

The final answer is the combination of both solutions.

$x = \frac{8 + 4 i \sqrt{3}}{7}, \frac{8 - 4 i \sqrt{3}}{7}$