Precalculus Examples

$3 x^{2} = 9 x - 11$

Step 1

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

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Step 1.1

Subtract $9 x$ from both sides of the equation.

$3 x^{2} - 9 x = - 11$

Step 1.2

Add $11$ to both sides of the equation.

$3 x^{2} - 9 x + 11 = 0$

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 $a = 3$ , $b = - 9$ , and $c = 11$ into the quadratic formula and solve for $x$ .

$\frac{9 \pm \sqrt{{(- 9)}^{2} - 4 \cdot (3 \cdot 11)}}{2 \cdot 3}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 9$ to the power of $2$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 3 \cdot 11}}{2 \cdot 3}$

Step 4.1.2

Multiply $- 4 \cdot 3 \cdot 11$ .

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Step 4.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{9 \pm \sqrt{81 - 12 \cdot 11}}{2 \cdot 3}$

Step 4.1.2.2

Multiply $- 12$ by $11$ .

$x = \frac{9 \pm \sqrt{81 - 132}}{2 \cdot 3}$

Step 4.1.3

Subtract $132$ from $81$ .

$x = \frac{9 \pm \sqrt{- 51}}{2 \cdot 3}$

Step 4.1.4

Rewrite $- 51$ as $- 1 (51)$ .

$x = \frac{9 \pm \sqrt{- 1 \cdot 51}}{2 \cdot 3}$

Step 4.1.5

Rewrite $\sqrt{- 1 (51)}$ as $\sqrt{- 1} \cdot \sqrt{51}$ .

$x = \frac{9 \pm \sqrt{- 1} \cdot \sqrt{51}}{2 \cdot 3}$

Step 4.1.6

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

$x = \frac{9 \pm i \sqrt{51}}{2 \cdot 3}$

Step 4.2

Multiply $2$ by $3$ .

$x = \frac{9 \pm i \sqrt{51}}{6}$