Precalculus Examples

$\frac{5 x^{2}}{4} = 2 (x - 1)$

Step 1

Multiply both sides by $4$ .

$\frac{5 x^{2}}{4} \cdot 4 = 2 (x - 1) \cdot 4$

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 $4$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{5 x^{2}}{4} \cdot 4 = 2 (x - 1) \cdot 4$

Step 2.1.1.2

Rewrite the expression.

$5 x^{2} = 2 (x - 1) \cdot 4$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $2 (x - 1) \cdot 4$ .

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Step 2.2.1.1

Apply the distributive property.

$5 x^{2} = (2 x + 2 \cdot - 1) \cdot 4$

Step 2.2.1.2

Multiply $2$ by $- 1$ .

$5 x^{2} = (2 x - 2) \cdot 4$

Step 2.2.1.3

Apply the distributive property.

$5 x^{2} = 2 x \cdot 4 - 2 \cdot 4$

Step 2.2.1.4

Multiply.

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Step 2.2.1.4.1

Multiply $4$ by $2$ .

$5 x^{2} = 8 x - 2 \cdot 4$

Step 2.2.1.4.2

Multiply $- 2$ by $4$ .

$5 x^{2} = 8 x - 8$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $8 x$ from both sides of the equation.

$5 x^{2} - 8 x = - 8$

Step 3.2

Add $8$ to both sides of the equation.

$5 x^{2} - 8 x + 8 = 0$

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 $a = 5$ , $b = - 8$ , and $c = 8$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (5 \cdot 8)}}{2 \cdot 5}$

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

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 5 \cdot 8}}{2 \cdot 5}$

Step 3.5.1.2

Multiply $- 4 \cdot 5 \cdot 8$ .

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Step 3.5.1.2.1

Multiply $- 4$ by $5$ .

$x = \frac{8 \pm \sqrt{64 - 20 \cdot 8}}{2 \cdot 5}$

Step 3.5.1.2.2

Multiply $- 20$ by $8$ .

$x = \frac{8 \pm \sqrt{64 - 160}}{2 \cdot 5}$

Step 3.5.1.3

Subtract $160$ from $64$ .

$x = \frac{8 \pm \sqrt{- 96}}{2 \cdot 5}$

Step 3.5.1.4

Rewrite $- 96$ as $- 1 (96)$ .

$x = \frac{8 \pm \sqrt{- 1 \cdot 96}}{2 \cdot 5}$

Step 3.5.1.5

Rewrite $\sqrt{- 1 (96)}$ as $\sqrt{- 1} \cdot \sqrt{96}$ .

$x = \frac{8 \pm \sqrt{- 1} \cdot \sqrt{96}}{2 \cdot 5}$

Step 3.5.1.6

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

$x = \frac{8 \pm i \cdot \sqrt{96}}{2 \cdot 5}$

Step 3.5.1.7

Rewrite $96$ as $4^{2} \cdot 6$ .

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Step 3.5.1.7.1

Factor $16$ out of $96$ .

$x = \frac{8 \pm i \cdot \sqrt{16 (6)}}{2 \cdot 5}$

Step 3.5.1.7.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{8 \pm i \cdot \sqrt{4^{2} \cdot 6}}{2 \cdot 5}$

Step 3.5.1.8

Pull terms out from under the radical.

$x = \frac{8 \pm i \cdot (4 \sqrt{6})}{2 \cdot 5}$

Step 3.5.1.9

Move $4$ to the left of $i$ .

$x = \frac{8 \pm 4 i \sqrt{6}}{2 \cdot 5}$

Step 3.5.2

Multiply $2$ by $5$ .

$x = \frac{8 \pm 4 i \sqrt{6}}{10}$

Step 3.5.3

Simplify $\frac{8 \pm 4 i \sqrt{6}}{10}$ .

$x = \frac{4 \pm 2 i \sqrt{6}}{5}$

Step 3.6

The final answer is the combination of both solutions.

$x = \frac{4 + 2 i \sqrt{6}}{5}, \frac{4 - 2 i \sqrt{6}}{5}$