Pre-Algebra Examples

$9 x^{2} - 1 - 10 x = 15 x^{2}$

Step 1

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

Tap for more steps...

Step 1.1

Subtract $15 x^{2}$ from both sides of the equation.

$9 x^{2} - 1 - 10 x - 15 x^{2} = 0$

Step 1.2

Subtract $15 x^{2}$ from $9 x^{2}$ .

$- 6 x^{2} - 1 - 10 x = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = - 6$ , $b = - 10$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (- 6 \cdot - 1)}}{2 \cdot - 6}$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Simplify the numerator.

Tap for more steps...

Step 4.1.1

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot - 6 \cdot - 1}}{2 \cdot - 6}$

Step 4.1.2

Multiply $- 4 \cdot - 6 \cdot - 1$ .

Tap for more steps...

Step 4.1.2.1

Multiply $- 4$ by $- 6$ .

$x = \frac{10 \pm \sqrt{100 + 24 \cdot - 1}}{2 \cdot - 6}$

Step 4.1.2.2

Multiply $24$ by $- 1$ .

$x = \frac{10 \pm \sqrt{100 - 24}}{2 \cdot - 6}$

Step 4.1.3

Subtract $24$ from $100$ .

$x = \frac{10 \pm \sqrt{76}}{2 \cdot - 6}$

Step 4.1.4

Rewrite $76$ as $2^{2} \cdot 19$ .

Tap for more steps...

Step 4.1.4.1

Factor $4$ out of $76$ .

$x = \frac{10 \pm \sqrt{4 (19)}}{2 \cdot - 6}$

Step 4.1.4.2

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

$x = \frac{10 \pm \sqrt{2^{2} \cdot 19}}{2 \cdot - 6}$

Step 4.1.5

Pull terms out from under the radical.

$x = \frac{10 \pm 2 \sqrt{19}}{2 \cdot - 6}$

Step 4.2

Multiply $2$ by $- 6$ .

$x = \frac{10 \pm 2 \sqrt{19}}{- 12}$

Step 4.3

Simplify $\frac{10 \pm 2 \sqrt{19}}{- 12}$ .

$x = \frac{5 \pm \sqrt{19}}{- 6}$

Step 4.4

Move the negative in front of the fraction.

$x = - \frac{5 \pm \sqrt{19}}{6}$

Step 5

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

Tap for more steps...

Step 5.1

Simplify the numerator.

Tap for more steps...

Step 5.1.1

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot - 6 \cdot - 1}}{2 \cdot - 6}$

Step 5.1.2

Multiply $- 4 \cdot - 6 \cdot - 1$ .

Tap for more steps...

Step 5.1.2.1

Multiply $- 4$ by $- 6$ .

$x = \frac{10 \pm \sqrt{100 + 24 \cdot - 1}}{2 \cdot - 6}$

Step 5.1.2.2

Multiply $24$ by $- 1$ .

$x = \frac{10 \pm \sqrt{100 - 24}}{2 \cdot - 6}$

Step 5.1.3

Subtract $24$ from $100$ .

$x = \frac{10 \pm \sqrt{76}}{2 \cdot - 6}$

Step 5.1.4

Rewrite $76$ as $2^{2} \cdot 19$ .

Tap for more steps...

Step 5.1.4.1

Factor $4$ out of $76$ .

$x = \frac{10 \pm \sqrt{4 (19)}}{2 \cdot - 6}$

Step 5.1.4.2

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

$x = \frac{10 \pm \sqrt{2^{2} \cdot 19}}{2 \cdot - 6}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{10 \pm 2 \sqrt{19}}{2 \cdot - 6}$

Step 5.2

Multiply $2$ by $- 6$ .

$x = \frac{10 \pm 2 \sqrt{19}}{- 12}$

Step 5.3

Simplify $\frac{10 \pm 2 \sqrt{19}}{- 12}$ .

$x = \frac{5 \pm \sqrt{19}}{- 6}$

Step 5.4

Move the negative in front of the fraction.

$x = - \frac{5 \pm \sqrt{19}}{6}$

Step 5.5

Change the $\pm$ to $+$ .

$x = - \frac{5 + \sqrt{19}}{6}$

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 $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot - 6 \cdot - 1}}{2 \cdot - 6}$

Step 6.1.2

Multiply $- 4 \cdot - 6 \cdot - 1$ .

Tap for more steps...

Step 6.1.2.1

Multiply $- 4$ by $- 6$ .

$x = \frac{10 \pm \sqrt{100 + 24 \cdot - 1}}{2 \cdot - 6}$

Step 6.1.2.2

Multiply $24$ by $- 1$ .

$x = \frac{10 \pm \sqrt{100 - 24}}{2 \cdot - 6}$

Step 6.1.3

Subtract $24$ from $100$ .

$x = \frac{10 \pm \sqrt{76}}{2 \cdot - 6}$

Step 6.1.4

Rewrite $76$ as $2^{2} \cdot 19$ .

Tap for more steps...

Step 6.1.4.1

Factor $4$ out of $76$ .

$x = \frac{10 \pm \sqrt{4 (19)}}{2 \cdot - 6}$

Step 6.1.4.2

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

$x = \frac{10 \pm \sqrt{2^{2} \cdot 19}}{2 \cdot - 6}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{10 \pm 2 \sqrt{19}}{2 \cdot - 6}$

Step 6.2

Multiply $2$ by $- 6$ .

$x = \frac{10 \pm 2 \sqrt{19}}{- 12}$

Step 6.3

Simplify $\frac{10 \pm 2 \sqrt{19}}{- 12}$ .

$x = \frac{5 \pm \sqrt{19}}{- 6}$

Step 6.4

Move the negative in front of the fraction.

$x = - \frac{5 \pm \sqrt{19}}{6}$

Step 6.5

Change the $\pm$ to $-$ .

$x = - \frac{5 - \sqrt{19}}{6}$

Step 7

The final answer is the combination of both solutions.

$x = - \frac{5 + \sqrt{19}}{6}, - \frac{5 - \sqrt{19}}{6}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{5 + \sqrt{19}}{6}, - \frac{5 - \sqrt{19}}{6}$

Decimal Form:

$x = - 1.55981649 \dots, - 0.10685017 \dots$