Precalculus Examples

$4 x^{2} + 5 x + \frac{13}{8} = 0$

Step 1

Multiply through by the least common denominator $8$ , then simplify.

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Step 1.1

Apply the distributive property.

$8 (4 x^{2}) + 8 (5 x) + 8 (\frac{13}{8}) = 0$

Step 1.2

Simplify.

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Step 1.2.1

Multiply $4$ by $8$ .

$32 x^{2} + 8 (5 x) + 8 (\frac{13}{8}) = 0$

Step 1.2.2

Multiply $5$ by $8$ .

$32 x^{2} + 40 x + 8 (\frac{13}{8}) = 0$

Step 1.2.3

Cancel the common factor of $8$ .

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Step 1.2.3.1

Cancel the common factor.

$32 x^{2} + 40 x + 8 (\frac{13}{8}) = 0$

Step 1.2.3.2

Rewrite the expression.

$32 x^{2} + 40 x + 13 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 32$ , $b = 40$ , and $c = 13$ into the quadratic formula and solve for $x$ .

$\frac{- 40 \pm \sqrt{40^{2} - 4 \cdot (32 \cdot 13)}}{2 \cdot 32}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $40$ to the power of $2$ .

$x = \frac{- 40 \pm \sqrt{1600 - 4 \cdot 32 \cdot 13}}{2 \cdot 32}$

Step 4.1.2

Multiply $- 4 \cdot 32 \cdot 13$ .

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Step 4.1.2.1

Multiply $- 4$ by $32$ .

$x = \frac{- 40 \pm \sqrt{1600 - 128 \cdot 13}}{2 \cdot 32}$

Step 4.1.2.2

Multiply $- 128$ by $13$ .

$x = \frac{- 40 \pm \sqrt{1600 - 1664}}{2 \cdot 32}$

Step 4.1.3

Subtract $1664$ from $1600$ .

$x = \frac{- 40 \pm \sqrt{- 64}}{2 \cdot 32}$

Step 4.1.4

Rewrite $- 64$ as $- 1 (64)$ .

$x = \frac{- 40 \pm \sqrt{- 1 \cdot 64}}{2 \cdot 32}$

Step 4.1.5

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

$x = \frac{- 40 \pm \sqrt{- 1} \cdot \sqrt{64}}{2 \cdot 32}$

Step 4.1.6

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

$x = \frac{- 40 \pm i \cdot \sqrt{64}}{2 \cdot 32}$

Step 4.1.7

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

$x = \frac{- 40 \pm i \cdot \sqrt{8^{2}}}{2 \cdot 32}$

Step 4.1.8

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

$x = \frac{- 40 \pm i \cdot 8}{2 \cdot 32}$

Step 4.1.9

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

$x = \frac{- 40 \pm 8 i}{2 \cdot 32}$

Step 4.2

Multiply $2$ by $32$ .

$x = \frac{- 40 \pm 8 i}{64}$

Step 4.3

Simplify $\frac{- 40 \pm 8 i}{64}$ .

$x = \frac{- 5 \pm i}{8}$