Precalculus Examples

$3 x^{2} = 8 x - 7$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

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

$3 x^{2} - 8 x = - 7$

Step 1.2

Add $7$ to both sides of the equation.

$3 x^{2} - 8 x + 7 = 0$

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 $a = 3$ , $b = - 8$ , and $c = 7$ into the quadratic formula and solve for $x$ .

$\frac{8 \pm \sqrt{{(- 8)}^{2} - 4 \cdot (3 \cdot 7)}}{2 \cdot 3}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$x = \frac{8 \pm \sqrt{64 - 4 \cdot 3 \cdot 7}}{2 \cdot 3}$

Step 4.1.2

Multiply $- 4 \cdot 3 \cdot 7$ .

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Step 4.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{8 \pm \sqrt{64 - 12 \cdot 7}}{2 \cdot 3}$

Step 4.1.2.2

Multiply $- 12$ by $7$ .

$x = \frac{8 \pm \sqrt{64 - 84}}{2 \cdot 3}$

Step 4.1.3

Subtract $84$ from $64$ .

$x = \frac{8 \pm \sqrt{- 20}}{2 \cdot 3}$

Step 4.1.4

Rewrite $- 20$ as $- 1 (20)$ .

$x = \frac{8 \pm \sqrt{- 1 \cdot 20}}{2 \cdot 3}$

Step 4.1.5

Rewrite $\sqrt{- 1 (20)}$ as $\sqrt{- 1} \cdot \sqrt{20}$ .

$x = \frac{8 \pm \sqrt{- 1} \cdot \sqrt{20}}{2 \cdot 3}$

Step 4.1.6

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

$x = \frac{8 \pm i \cdot \sqrt{20}}{2 \cdot 3}$

Step 4.1.7

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 4.1.7.1

Factor $4$ out of $20$ .

$x = \frac{8 \pm i \cdot \sqrt{4 (5)}}{2 \cdot 3}$

Step 4.1.7.2

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

$x = \frac{8 \pm i \cdot \sqrt{2^{2} \cdot 5}}{2 \cdot 3}$

Step 4.1.8

Pull terms out from under the radical.

$x = \frac{8 \pm i \cdot (2 \sqrt{5})}{2 \cdot 3}$

Step 4.1.9

Move $2$ to the left of $i$ .

$x = \frac{8 \pm 2 i \sqrt{5}}{2 \cdot 3}$

Step 4.2

Multiply $2$ by $3$ .

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

Step 4.3

Simplify $\frac{8 \pm 2 i \sqrt{5}}{6}$ .

$x = \frac{4 \pm i \sqrt{5}}{3}$