Pre-Algebra Examples

$- 5 y (2 y - 6) + 2 y = 6 (y + 3)$

Step 1

Simplify $- 5 y (2 y - 6) + 2 y$ .

Tap for more steps...

Step 1.1

Simplify each term.

Tap for more steps...

Step 1.1.1

Apply the distributive property.

$- 5 y (2 y) - 5 y \cdot - 6 + 2 y = 6 (y + 3)$

Step 1.1.2

Rewrite using the commutative property of multiplication.

$- 5 \cdot 2 y \cdot y - 5 y \cdot - 6 + 2 y = 6 (y + 3)$

Step 1.1.3

Multiply $- 6$ by $- 5$ .

$- 5 \cdot 2 y \cdot y + 30 y + 2 y = 6 (y + 3)$

Step 1.1.4

Simplify each term.

Tap for more steps...

Step 1.1.4.1

Multiply $y$ by $y$ by adding the exponents.

Tap for more steps...

Step 1.1.4.1.1

Move $y$ .

$- 5 \cdot 2 (y \cdot y) + 30 y + 2 y = 6 (y + 3)$

Step 1.1.4.1.2

Multiply $y$ by $y$ .

$- 5 \cdot 2 y^{2} + 30 y + 2 y = 6 (y + 3)$

Step 1.1.4.2

Multiply $- 5$ by $2$ .

$- 10 y^{2} + 30 y + 2 y = 6 (y + 3)$

Step 1.2

Add $30 y$ and $2 y$ .

$- 10 y^{2} + 32 y = 6 (y + 3)$

Step 2

Simplify $6 (y + 3)$ .

Tap for more steps...

Step 2.1

Apply the distributive property.

$- 10 y^{2} + 32 y = 6 y + 6 \cdot 3$

Step 2.2

Multiply $6$ by $3$ .

$- 10 y^{2} + 32 y = 6 y + 18$

Step 3

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

Tap for more steps...

Step 3.1

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

$- 10 y^{2} + 32 y - 6 y = 18$

Step 3.2

Subtract $6 y$ from $32 y$ .

$- 10 y^{2} + 26 y = 18$

Step 4

Subtract $18$ from both sides of the equation.

$- 10 y^{2} + 26 y - 18 = 0$

Step 5

Factor $- 2$ out of $- 10 y^{2} + 26 y - 18$ .

Tap for more steps...

Step 5.1

Factor $- 2$ out of $- 10 y^{2}$ .

$- 2 (5 y^{2}) + 26 y - 18 = 0$

Step 5.2

Factor $- 2$ out of $26 y$ .

$- 2 (5 y^{2}) - 2 (- 13 y) - 18 = 0$

Step 5.3

Factor $- 2$ out of $- 18$ .

$- 2 (5 y^{2}) - 2 (- 13 y) - 2 \cdot 9 = 0$

Step 5.4

Factor $- 2$ out of $- 2 (5 y^{2}) - 2 (- 13 y)$ .

$- 2 (5 y^{2} - 13 y) - 2 \cdot 9 = 0$

Step 5.5

Factor $- 2$ out of $- 2 (5 y^{2} - 13 y) - 2 (9)$ .

$- 2 (5 y^{2} - 13 y + 9) = 0$

Step 6

Divide each term in $- 2 (5 y^{2} - 13 y + 9) = 0$ by $- 2$ and simplify.

Tap for more steps...

Step 6.1

Divide each term in $- 2 (5 y^{2} - 13 y + 9) = 0$ by $- 2$ .

$\frac{- 2 (5 y^{2} - 13 y + 9)}{- 2} = \frac{0}{- 2}$

Step 6.2

Simplify the left side.

Tap for more steps...

Step 6.2.1

Cancel the common factor of $- 2$ .

Tap for more steps...

Step 6.2.1.1

Cancel the common factor.

$\frac{- 2 (5 y^{2} - 13 y + 9)}{- 2} = \frac{0}{- 2}$

Step 6.2.1.2

Divide $5 y^{2} - 13 y + 9$ by $1$ .

$5 y^{2} - 13 y + 9 = \frac{0}{- 2}$

Step 6.3

Simplify the right side.

Tap for more steps...

Step 6.3.1

Divide $0$ by $- 2$ .

$5 y^{2} - 13 y + 9 = 0$

Step 7

Use the quadratic formula to find the solutions.

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

Step 8

Substitute the values $a = 5$ , $b = - 13$ , and $c = 9$ into the quadratic formula and solve for $y$ .

$\frac{13 \pm \sqrt{{(- 13)}^{2} - 4 \cdot (5 \cdot 9)}}{2 \cdot 5}$

Step 9

Simplify.

Tap for more steps...

Step 9.1

Simplify the numerator.

Tap for more steps...

Step 9.1.1

Raise $- 13$ to the power of $2$ .

$y = \frac{13 \pm \sqrt{169 - 4 \cdot 5 \cdot 9}}{2 \cdot 5}$

Step 9.1.2

Multiply $- 4 \cdot 5 \cdot 9$ .

Tap for more steps...

Step 9.1.2.1

Multiply $- 4$ by $5$ .

$y = \frac{13 \pm \sqrt{169 - 20 \cdot 9}}{2 \cdot 5}$

Step 9.1.2.2

Multiply $- 20$ by $9$ .

$y = \frac{13 \pm \sqrt{169 - 180}}{2 \cdot 5}$

Step 9.1.3

Subtract $180$ from $169$ .

$y = \frac{13 \pm \sqrt{- 11}}{2 \cdot 5}$

Step 9.1.4

Rewrite $- 11$ as $- 1 (11)$ .

$y = \frac{13 \pm \sqrt{- 1 \cdot 11}}{2 \cdot 5}$

Step 9.1.5

Rewrite $\sqrt{- 1 (11)}$ as $\sqrt{- 1} \cdot \sqrt{11}$ .

$y = \frac{13 \pm \sqrt{- 1} \cdot \sqrt{11}}{2 \cdot 5}$

Step 9.1.6

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

$y = \frac{13 \pm i \sqrt{11}}{2 \cdot 5}$

Step 9.2

Multiply $2$ by $5$ .

$y = \frac{13 \pm i \sqrt{11}}{10}$

Step 10

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

Tap for more steps...

Step 10.1

Simplify the numerator.

Tap for more steps...

Step 10.1.1

Raise $- 13$ to the power of $2$ .

$y = \frac{13 \pm \sqrt{169 - 4 \cdot 5 \cdot 9}}{2 \cdot 5}$

Step 10.1.2

Multiply $- 4 \cdot 5 \cdot 9$ .

Tap for more steps...

Step 10.1.2.1

Multiply $- 4$ by $5$ .

$y = \frac{13 \pm \sqrt{169 - 20 \cdot 9}}{2 \cdot 5}$

Step 10.1.2.2

Multiply $- 20$ by $9$ .

$y = \frac{13 \pm \sqrt{169 - 180}}{2 \cdot 5}$

Step 10.1.3

Subtract $180$ from $169$ .

$y = \frac{13 \pm \sqrt{- 11}}{2 \cdot 5}$

Step 10.1.4

Rewrite $- 11$ as $- 1 (11)$ .

$y = \frac{13 \pm \sqrt{- 1 \cdot 11}}{2 \cdot 5}$

Step 10.1.5

Rewrite $\sqrt{- 1 (11)}$ as $\sqrt{- 1} \cdot \sqrt{11}$ .

$y = \frac{13 \pm \sqrt{- 1} \cdot \sqrt{11}}{2 \cdot 5}$

Step 10.1.6

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

$y = \frac{13 \pm i \sqrt{11}}{2 \cdot 5}$

Step 10.2

Multiply $2$ by $5$ .

$y = \frac{13 \pm i \sqrt{11}}{10}$

Step 10.3

Change the $\pm$ to $+$ .

$y = \frac{13 + i \sqrt{11}}{10}$

Step 11

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

Tap for more steps...

Step 11.1

Simplify the numerator.

Tap for more steps...

Step 11.1.1

Raise $- 13$ to the power of $2$ .

$y = \frac{13 \pm \sqrt{169 - 4 \cdot 5 \cdot 9}}{2 \cdot 5}$

Step 11.1.2

Multiply $- 4 \cdot 5 \cdot 9$ .

Tap for more steps...

Step 11.1.2.1

Multiply $- 4$ by $5$ .

$y = \frac{13 \pm \sqrt{169 - 20 \cdot 9}}{2 \cdot 5}$

Step 11.1.2.2

Multiply $- 20$ by $9$ .

$y = \frac{13 \pm \sqrt{169 - 180}}{2 \cdot 5}$

Step 11.1.3

Subtract $180$ from $169$ .

$y = \frac{13 \pm \sqrt{- 11}}{2 \cdot 5}$

Step 11.1.4

Rewrite $- 11$ as $- 1 (11)$ .

$y = \frac{13 \pm \sqrt{- 1 \cdot 11}}{2 \cdot 5}$

Step 11.1.5

Rewrite $\sqrt{- 1 (11)}$ as $\sqrt{- 1} \cdot \sqrt{11}$ .

$y = \frac{13 \pm \sqrt{- 1} \cdot \sqrt{11}}{2 \cdot 5}$

Step 11.1.6

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

$y = \frac{13 \pm i \sqrt{11}}{2 \cdot 5}$

Step 11.2

Multiply $2$ by $5$ .

$y = \frac{13 \pm i \sqrt{11}}{10}$

Step 11.3

Change the $\pm$ to $-$ .

$y = \frac{13 - i \sqrt{11}}{10}$

Step 12

The final answer is the combination of both solutions.

$y = \frac{13 + i \sqrt{11}}{10}, \frac{13 - i \sqrt{11}}{10}$