Algebra Examples

$x^{2} + 3 x + 8 = 6 x - 7$

Step 1

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

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Step 1.1

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

$x^{2} + 3 x + 8 - 6 x = - 7$

Step 1.2

Subtract $6 x$ from $3 x$ .

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

Step 2

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

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Step 2.1

Add $7$ to both sides of the equation.

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

Step 2.2

Add $8$ and $7$ .

$x^{2} - 3 x + 15 = 0$

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 $a = 1$ , $b = - 3$ , and $c = 15$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (1 \cdot 15)}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot 15}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 15$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 15}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $15$ .

$x = \frac{3 \pm \sqrt{9 - 60}}{2 \cdot 1}$

Step 5.1.3

Subtract $60$ from $9$ .

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

Step 5.1.4

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

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

Step 5.1.5

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

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

Step 5.1.6

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

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

Step 5.2

Multiply $2$ by $1$ .

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

Step 6

The final answer is the combination of both solutions.

$x = \frac{3 + i \sqrt{51}}{2}, \frac{3 - i \sqrt{51}}{2}$