Precalculus Examples

$90 x^{2} + 1 = 6 x$

Step 1

Subtract $6 x$ from both sides of the equation.

$90 x^{2} + 1 - 6 x = 0$

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 $a = 90$ , $b = - 6$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (90 \cdot 1)}}{2 \cdot 90}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 90 \cdot 1}}{2 \cdot 90}$

Step 4.1.2

Multiply $- 4 \cdot 90 \cdot 1$ .

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Step 4.1.2.1

Multiply $- 4$ by $90$ .

$x = \frac{6 \pm \sqrt{36 - 360 \cdot 1}}{2 \cdot 90}$

Step 4.1.2.2

Multiply $- 360$ by $1$ .

$x = \frac{6 \pm \sqrt{36 - 360}}{2 \cdot 90}$

Step 4.1.3

Subtract $360$ from $36$ .

$x = \frac{6 \pm \sqrt{- 324}}{2 \cdot 90}$

Step 4.1.4

Rewrite $- 324$ as $- 1 (324)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 324}}{2 \cdot 90}$

Step 4.1.5

Rewrite $\sqrt{- 1 (324)}$ as $\sqrt{- 1} \cdot \sqrt{324}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{324}}{2 \cdot 90}$

Step 4.1.6

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

$x = \frac{6 \pm i \cdot \sqrt{324}}{2 \cdot 90}$

Step 4.1.7

Rewrite $324$ as $18^{2}$ .

$x = \frac{6 \pm i \cdot \sqrt{18^{2}}}{2 \cdot 90}$

Step 4.1.8

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

$x = \frac{6 \pm i \cdot 18}{2 \cdot 90}$

Step 4.1.9

Move $18$ to the left of $i$ .

$x = \frac{6 \pm 18 i}{2 \cdot 90}$

Step 4.2

Multiply $2$ by $90$ .

$x = \frac{6 \pm 18 i}{180}$

Step 4.3

Simplify $\frac{6 \pm 18 i}{180}$ .

$x = \frac{1 \pm 3 i}{30}$