Precalculus Examples

$x^{2} + x + 11 = 5 x - 8$

Step 1

Move all terms containing $x$ to the left side of the equation.

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Step 1.1

Subtract $5 x$ from both sides of the equation.

$x^{2} + x + 11 - 5 x = - 8$

Step 1.2

Subtract $5 x$ from $x$ .

$x^{2} - 4 x + 11 = - 8$

Step 2

Move all terms to the left side of the equation and simplify.

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Step 2.1

Add $8$ to both sides of the equation.

$x^{2} - 4 x + 11 + 8 = 0$

Step 2.2

Add $11$ and $8$ .

$x^{2} - 4 x + 19 = 0$

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 $a = 1$ , $b = - 4$ , and $c = 19$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot 19)}}{2 \cdot 1}$

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 $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 19}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 19$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 19}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $19$ .

$x = \frac{4 \pm \sqrt{16 - 76}}{2 \cdot 1}$

Step 5.1.3

Subtract $76$ from $16$ .

$x = \frac{4 \pm \sqrt{- 60}}{2 \cdot 1}$

Step 5.1.4

Rewrite $- 60$ as $- 1 (60)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 60}}{2 \cdot 1}$

Step 5.1.5

Rewrite $\sqrt{- 1 (60)}$ as $\sqrt{- 1} \cdot \sqrt{60}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{60}}{2 \cdot 1}$

Step 5.1.6

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

$x = \frac{4 \pm i \cdot \sqrt{60}}{2 \cdot 1}$

Step 5.1.7

Rewrite $60$ as $2^{2} \cdot 15$ .

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Step 5.1.7.1

Factor $4$ out of $60$ .

$x = \frac{4 \pm i \cdot \sqrt{4 (15)}}{2 \cdot 1}$

Step 5.1.7.2

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

$x = \frac{4 \pm i \cdot \sqrt{2^{2} \cdot 15}}{2 \cdot 1}$

Step 5.1.8

Pull terms out from under the radical.

$x = \frac{4 \pm i \cdot (2 \sqrt{15})}{2 \cdot 1}$

Step 5.1.9

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

$x = \frac{4 \pm 2 i \sqrt{15}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 i \sqrt{15}}{2}$

Step 5.3

Simplify $\frac{4 \pm 2 i \sqrt{15}}{2}$ .

$x = 2 \pm i \sqrt{15}$

Step 6

The final answer is the combination of both solutions.

$x = 2 + i \sqrt{15}, 2 - i \sqrt{15}$