Precalculus Examples

$3 x^{2} = 4 x - 10$

Step 1

Subtract $4 x$ from both sides of the equation.

$3 x^{2} - 4 x = - 10$

Step 2

Add $10$ to both sides of the equation.

$3 x^{2} - 4 x + 10 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 3$ , $b = - 4$ , and $c = 10$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (3 \cdot 10)}}{2 \cdot 3}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 3 \cdot 10}}{2 \cdot 3}$

Step 5.1.2

Multiply $- 4 \cdot 3 \cdot 10$ .

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Step 5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{4 \pm \sqrt{16 - 12 \cdot 10}}{2 \cdot 3}$

Step 5.1.2.2

Multiply $- 12$ by $10$ .

$x = \frac{4 \pm \sqrt{16 - 120}}{2 \cdot 3}$

Step 5.1.3

Subtract $120$ from $16$ .

$x = \frac{4 \pm \sqrt{- 104}}{2 \cdot 3}$

Step 5.1.4

Rewrite $- 104$ as $- 1 (104)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 104}}{2 \cdot 3}$

Step 5.1.5

Rewrite $\sqrt{- 1 (104)}$ as $\sqrt{- 1} \cdot \sqrt{104}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{104}}{2 \cdot 3}$

Step 5.1.6

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

$x = \frac{4 \pm i \cdot \sqrt{104}}{2 \cdot 3}$

Step 5.1.7

Rewrite $104$ as $2^{2} \cdot 26$ .

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Step 5.1.7.1

Factor $4$ out of $104$ .

$x = \frac{4 \pm i \cdot \sqrt{4 (26)}}{2 \cdot 3}$

Step 5.1.7.2

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

$x = \frac{4 \pm i \cdot \sqrt{2^{2} \cdot 26}}{2 \cdot 3}$

Step 5.1.8

Pull terms out from under the radical.

$x = \frac{4 \pm i \cdot (2 \sqrt{26})}{2 \cdot 3}$

Step 5.1.9

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

$x = \frac{4 \pm 2 i \sqrt{26}}{2 \cdot 3}$

Step 5.2

Multiply $2$ by $3$ .

$x = \frac{4 \pm 2 i \sqrt{26}}{6}$

Step 5.3

Simplify $\frac{4 \pm 2 i \sqrt{26}}{6}$ .

$x = \frac{2 \pm i \sqrt{26}}{3}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{2 + i \sqrt{26}}{3}, \frac{2 - i \sqrt{26}}{3}$