Pre-Algebra Examples

$4 x^{2} + 5 x = 2 x + 2$

Step 1

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

Tap for more steps...

Step 1.1

Subtract $2 x$ from both sides of the equation.

$4 x^{2} + 5 x - 2 x = 2$

Step 1.2

Subtract $2 x$ from $5 x$ .

$4 x^{2} + 3 x = 2$

Step 2

Subtract $2$ from both sides of the equation.

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

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 $a = 4$ , $b = 3$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

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

Step 5

Simplify.

Tap for more steps...

Step 5.1

Simplify the numerator.

Tap for more steps...

Step 5.1.1

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

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

Step 5.1.2

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

Tap for more steps...

Step 5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{- 3 \pm \sqrt{9 - 16 \cdot - 2}}{2 \cdot 4}$

Step 5.1.2.2

Multiply $- 16$ by $- 2$ .

$x = \frac{- 3 \pm \sqrt{9 + 32}}{2 \cdot 4}$

Step 5.1.3

Add $9$ and $32$ .

$x = \frac{- 3 \pm \sqrt{41}}{2 \cdot 4}$

Step 5.2

Multiply $2$ by $4$ .

$x = \frac{- 3 \pm \sqrt{41}}{8}$

Step 6

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

Tap for more steps...

Step 6.1

Simplify the numerator.

Tap for more steps...

Step 6.1.1

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

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

Step 6.1.2

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

Tap for more steps...

Step 6.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{- 3 \pm \sqrt{9 - 16 \cdot - 2}}{2 \cdot 4}$

Step 6.1.2.2

Multiply $- 16$ by $- 2$ .

$x = \frac{- 3 \pm \sqrt{9 + 32}}{2 \cdot 4}$

Step 6.1.3

Add $9$ and $32$ .

$x = \frac{- 3 \pm \sqrt{41}}{2 \cdot 4}$

Step 6.2

Multiply $2$ by $4$ .

$x = \frac{- 3 \pm \sqrt{41}}{8}$

Step 6.3

Change the $\pm$ to $+$ .

$x = \frac{- 3 + \sqrt{41}}{8}$

Step 6.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + \sqrt{41}}{8}$

Step 6.5

Factor $- 1$ out of $\sqrt{41}$ .

$x = \frac{- 1 \cdot 3 - 1 (- \sqrt{41})}{8}$

Step 6.6

Factor $- 1$ out of $- 1 (3) - 1 (- \sqrt{41})$ .

$x = \frac{- 1 (3 - \sqrt{41})}{8}$

Step 6.7

Move the negative in front of the fraction.

$x = - \frac{3 - \sqrt{41}}{8}$

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

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

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

Step 7.1.2

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

Tap for more steps...

Step 7.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{- 3 \pm \sqrt{9 - 16 \cdot - 2}}{2 \cdot 4}$

Step 7.1.2.2

Multiply $- 16$ by $- 2$ .

$x = \frac{- 3 \pm \sqrt{9 + 32}}{2 \cdot 4}$

Step 7.1.3

Add $9$ and $32$ .

$x = \frac{- 3 \pm \sqrt{41}}{2 \cdot 4}$

Step 7.2

Multiply $2$ by $4$ .

$x = \frac{- 3 \pm \sqrt{41}}{8}$

Step 7.3

Change the $\pm$ to $-$ .

$x = \frac{- 3 - \sqrt{41}}{8}$

Step 7.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - \sqrt{41}}{8}$

Step 7.5

Factor $- 1$ out of $- \sqrt{41}$ .

$x = \frac{- 1 \cdot 3 - (\sqrt{41})}{8}$

Step 7.6

Factor $- 1$ out of $- 1 (3) - (\sqrt{41})$ .

$x = \frac{- 1 (3 + \sqrt{41})}{8}$

Step 7.7

Move the negative in front of the fraction.

$x = - \frac{3 + \sqrt{41}}{8}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{3 - \sqrt{41}}{8}, - \frac{3 + \sqrt{41}}{8}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3 - \sqrt{41}}{8}, - \frac{3 + \sqrt{41}}{8}$

Decimal Form:

$x = 0.42539052 \dots, - 1.17539052 \dots$