Algebra Examples

$f (x) = 5 x^{2} + 7$ , $f (x) = 8 - 3 x$

Step 1

Substitute $8 - 3 x$ for $f (x)$ .

$8 - 3 x = 5 x^{2} + 7$

Step 2

Solve for $x$ .

Tap for more steps...

Step 2.1

Subtract $5 x^{2}$ from both sides of the equation.

$8 - 3 x - 5 x^{2} = 7$

Step 2.2

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

Tap for more steps...

Step 2.2.1

Subtract $7$ from both sides of the equation.

$8 - 3 x - 5 x^{2} - 7 = 0$

Step 2.2.2

Subtract $7$ from $8$ .

$- 3 x - 5 x^{2} + 1 = 0$

Step 2.3

Use the quadratic formula to find the solutions.

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

Step 2.4

Substitute the values $a = - 5$ , $b = - 3$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (- 5 \cdot 1)}}{2 \cdot - 5}$

Step 2.5

Simplify.

Tap for more steps...

Step 2.5.1

Simplify the numerator.

Tap for more steps...

Step 2.5.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 5 \cdot 1}}{2 \cdot - 5}$

Step 2.5.1.2

Multiply $- 4 \cdot - 5 \cdot 1$ .

Tap for more steps...

Step 2.5.1.2.1

Multiply $- 4$ by $- 5$ .

$x = \frac{3 \pm \sqrt{9 + 20 \cdot 1}}{2 \cdot - 5}$

Step 2.5.1.2.2

Multiply $20$ by $1$ .

$x = \frac{3 \pm \sqrt{9 + 20}}{2 \cdot - 5}$

Step 2.5.1.3

Add $9$ and $20$ .

$x = \frac{3 \pm \sqrt{29}}{2 \cdot - 5}$

Step 2.5.2

Multiply $2$ by $- 5$ .

$x = \frac{3 \pm \sqrt{29}}{- 10}$

Step 2.5.3

Move the negative in front of the fraction.

$x = - \frac{3 \pm \sqrt{29}}{10}$

Step 2.6

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

Tap for more steps...

Step 2.6.1

Simplify the numerator.

Tap for more steps...

Step 2.6.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 5 \cdot 1}}{2 \cdot - 5}$

Step 2.6.1.2

Multiply $- 4 \cdot - 5 \cdot 1$ .

Tap for more steps...

Step 2.6.1.2.1

Multiply $- 4$ by $- 5$ .

$x = \frac{3 \pm \sqrt{9 + 20 \cdot 1}}{2 \cdot - 5}$

Step 2.6.1.2.2

Multiply $20$ by $1$ .

$x = \frac{3 \pm \sqrt{9 + 20}}{2 \cdot - 5}$

Step 2.6.1.3

Add $9$ and $20$ .

$x = \frac{3 \pm \sqrt{29}}{2 \cdot - 5}$

Step 2.6.2

Multiply $2$ by $- 5$ .

$x = \frac{3 \pm \sqrt{29}}{- 10}$

Step 2.6.3

Move the negative in front of the fraction.

$x = - \frac{3 \pm \sqrt{29}}{10}$

Step 2.6.4

Change the $\pm$ to $+$ .

$x = - \frac{3 + \sqrt{29}}{10}$

Step 2.7

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

Tap for more steps...

Step 2.7.1

Simplify the numerator.

Tap for more steps...

Step 2.7.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 5 \cdot 1}}{2 \cdot - 5}$

Step 2.7.1.2

Multiply $- 4 \cdot - 5 \cdot 1$ .

Tap for more steps...

Step 2.7.1.2.1

Multiply $- 4$ by $- 5$ .

$x = \frac{3 \pm \sqrt{9 + 20 \cdot 1}}{2 \cdot - 5}$

Step 2.7.1.2.2

Multiply $20$ by $1$ .

$x = \frac{3 \pm \sqrt{9 + 20}}{2 \cdot - 5}$

Step 2.7.1.3

Add $9$ and $20$ .

$x = \frac{3 \pm \sqrt{29}}{2 \cdot - 5}$

Step 2.7.2

Multiply $2$ by $- 5$ .

$x = \frac{3 \pm \sqrt{29}}{- 10}$

Step 2.7.3

Move the negative in front of the fraction.

$x = - \frac{3 \pm \sqrt{29}}{10}$

Step 2.7.4

Change the $\pm$ to $-$ .

$x = - \frac{3 - \sqrt{29}}{10}$

Step 2.8

The final answer is the combination of both solutions.

$x = - \frac{3 + \sqrt{29}}{10}, - \frac{3 - \sqrt{29}}{10}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3 + \sqrt{29}}{10}, - \frac{3 - \sqrt{29}}{10}$

Decimal Form:

$x = - 0.83851648 \dots, 0.23851648 \dots$