Pre-Algebra Examples

$x^{2} - 6 x + 4 = - 8 x - 7 x^{2}$

Step 1

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

Tap for more steps...

Step 1.1

Add $8 x$ to both sides of the equation.

$x^{2} - 6 x + 4 + 8 x = - 7 x^{2}$

Step 1.2

Add $7 x^{2}$ to both sides of the equation.

$x^{2} - 6 x + 4 + 8 x + 7 x^{2} = 0$

Step 1.3

Add $x^{2}$ and $7 x^{2}$ .

$8 x^{2} - 6 x + 4 + 8 x = 0$

Step 1.4

Add $- 6 x$ and $8 x$ .

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

Step 2

Factor $2$ out of $8 x^{2} + 2 x + 4$ .

Tap for more steps...

Step 2.1

Factor $2$ out of $8 x^{2}$ .

$2 (4 x^{2}) + 2 x + 4 = 0$

Step 2.2

Factor $2$ out of $2 x$ .

$2 (4 x^{2}) + 2 (x) + 4 = 0$

Step 2.3

Factor $2$ out of $4$ .

$2 (4 x^{2}) + 2 x + 2 \cdot 2 = 0$

Step 2.4

Factor $2$ out of $2 (4 x^{2}) + 2 x$ .

$2 (4 x^{2} + x) + 2 \cdot 2 = 0$

Step 2.5

Factor $2$ out of $2 (4 x^{2} + x) + 2 \cdot 2$ .

$2 (4 x^{2} + x + 2) = 0$

Step 3

Divide each term in $2 (4 x^{2} + x + 2) = 0$ by $2$ and simplify.

Tap for more steps...

Step 3.1

Divide each term in $2 (4 x^{2} + x + 2) = 0$ by $2$ .

$\frac{2 (4 x^{2} + x + 2)}{2} = \frac{0}{2}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.1

Cancel the common factor.

$\frac{2 (4 x^{2} + x + 2)}{2} = \frac{0}{2}$

Step 3.2.1.2

Divide $4 x^{2} + x + 2$ by $1$ .

$4 x^{2} + x + 2 = \frac{0}{2}$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Divide $0$ by $2$ .

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

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 4$ , $b = 1$ , and $c = 2$ into the quadratic formula and solve for $x$ .

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

Step 6

Simplify.

Tap for more steps...

Step 6.1

Simplify the numerator.

Tap for more steps...

Step 6.1.1

One to any power is one.

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

Step 6.1.2

Multiply $- 4 \cdot 4 \cdot 2$ .

Tap for more steps...

Step 6.1.2.1

Multiply $- 4$ by $4$ .

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

Step 6.1.2.2

Multiply $- 16$ by $2$ .

$x = \frac{- 1 \pm \sqrt{1 - 32}}{2 \cdot 4}$

Step 6.1.3

Subtract $32$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 31}}{2 \cdot 4}$

Step 6.1.4

Rewrite $- 31$ as $- 1 (31)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 31}}{2 \cdot 4}$

Step 6.1.5

Rewrite $\sqrt{- 1 (31)}$ as $\sqrt{- 1} \cdot \sqrt{31}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{31}}{2 \cdot 4}$

Step 6.1.6

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

$x = \frac{- 1 \pm i \sqrt{31}}{2 \cdot 4}$

Step 6.2

Multiply $2$ by $4$ .

$x = \frac{- 1 \pm i \sqrt{31}}{8}$

Step 7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Tap for more steps...

Step 7.1

Simplify the numerator.

Tap for more steps...

Step 7.1.1

One to any power is one.

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

Step 7.1.2

Multiply $- 4 \cdot 4 \cdot 2$ .

Tap for more steps...

Step 7.1.2.1

Multiply $- 4$ by $4$ .

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

Step 7.1.2.2

Multiply $- 16$ by $2$ .

$x = \frac{- 1 \pm \sqrt{1 - 32}}{2 \cdot 4}$

Step 7.1.3

Subtract $32$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 31}}{2 \cdot 4}$

Step 7.1.4

Rewrite $- 31$ as $- 1 (31)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 31}}{2 \cdot 4}$

Step 7.1.5

Rewrite $\sqrt{- 1 (31)}$ as $\sqrt{- 1} \cdot \sqrt{31}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{31}}{2 \cdot 4}$

Step 7.1.6

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

$x = \frac{- 1 \pm i \sqrt{31}}{2 \cdot 4}$

Step 7.2

Multiply $2$ by $4$ .

$x = \frac{- 1 \pm i \sqrt{31}}{8}$

Step 7.3

Change the $\pm$ to $+$ .

$x = \frac{- 1 + i \sqrt{31}}{8}$

Step 7.4

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

$x = \frac{- 1 \cdot 1 + i \sqrt{31}}{8}$

Step 7.5

Factor $- 1$ out of $i \sqrt{31}$ .

$x = \frac{- 1 \cdot 1 - (- i \sqrt{31})}{8}$

Step 7.6

Factor $- 1$ out of $- 1 (1) - (- i \sqrt{31})$ .

$x = \frac{- 1 (1 - i \sqrt{31})}{8}$

Step 7.7

Move the negative in front of the fraction.

$x = - \frac{1 - i \sqrt{31}}{8}$

Step 8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Tap for more steps...

Step 8.1

Simplify the numerator.

Tap for more steps...

Step 8.1.1

One to any power is one.

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

Step 8.1.2

Multiply $- 4 \cdot 4 \cdot 2$ .

Tap for more steps...

Step 8.1.2.1

Multiply $- 4$ by $4$ .

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

Step 8.1.2.2

Multiply $- 16$ by $2$ .

$x = \frac{- 1 \pm \sqrt{1 - 32}}{2 \cdot 4}$

Step 8.1.3

Subtract $32$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 31}}{2 \cdot 4}$

Step 8.1.4

Rewrite $- 31$ as $- 1 (31)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 31}}{2 \cdot 4}$

Step 8.1.5

Rewrite $\sqrt{- 1 (31)}$ as $\sqrt{- 1} \cdot \sqrt{31}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{31}}{2 \cdot 4}$

Step 8.1.6

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

$x = \frac{- 1 \pm i \sqrt{31}}{2 \cdot 4}$

Step 8.2

Multiply $2$ by $4$ .

$x = \frac{- 1 \pm i \sqrt{31}}{8}$

Step 8.3

Change the $\pm$ to $-$ .

$x = \frac{- 1 - i \sqrt{31}}{8}$

Step 8.4

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

$x = \frac{- 1 \cdot 1 - i \sqrt{31}}{8}$

Step 8.5

Factor $- 1$ out of $- i \sqrt{31}$ .

$x = \frac{- 1 \cdot 1 - (i \sqrt{31})}{8}$

Step 8.6

Factor $- 1$ out of $- 1 (1) - (i \sqrt{31})$ .

$x = \frac{- 1 (1 + i \sqrt{31})}{8}$

Step 8.7

Move the negative in front of the fraction.

$x = - \frac{1 + i \sqrt{31}}{8}$

Step 9

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{31}}{8}, - \frac{1 + i \sqrt{31}}{8}$