Basic Math Examples

$15 y (6 + y) = 135 + 15 y$

Step 1

Simplify $15 y (6 + y)$ .

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Step 1.1

Rewrite.

$0 + 0 + 15 y (6 + y) = 135 + 15 y$

Step 1.2

Simplify by adding zeros.

$15 y (6 + y) = 135 + 15 y$

Step 1.3

Apply the distributive property.

$15 y \cdot 6 + 15 y \cdot y = 135 + 15 y$

Step 1.4

Multiply $6$ by $15$ .

$90 y + 15 y \cdot y = 135 + 15 y$

Step 1.5

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

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Step 1.5.1

Move $y$ .

$90 y + 15 (y \cdot y) = 135 + 15 y$

Step 1.5.2

Multiply $y$ by $y$ .

$90 y + 15 y^{2} = 135 + 15 y$

Step 2

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

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Step 2.1

Subtract $15 y$ from both sides of the equation.

$90 y + 15 y^{2} - 15 y = 135$

Step 2.2

Subtract $15 y$ from $90 y$ .

$15 y^{2} + 75 y = 135$

Step 3

Subtract $135$ from both sides of the equation.

$15 y^{2} + 75 y - 135 = 0$

Step 4

Factor $15$ out of $15 y^{2} + 75 y - 135$ .

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Step 4.1

Factor $15$ out of $15 y^{2}$ .

$15 (y^{2}) + 75 y - 135 = 0$

Step 4.2

Factor $15$ out of $75 y$ .

$15 (y^{2}) + 15 (5 y) - 135 = 0$

Step 4.3

Factor $15$ out of $- 135$ .

$15 y^{2} + 15 (5 y) + 15 \cdot - 9 = 0$

Step 4.4

Factor $15$ out of $15 y^{2} + 15 (5 y)$ .

$15 (y^{2} + 5 y) + 15 \cdot - 9 = 0$

Step 4.5

Factor $15$ out of $15 (y^{2} + 5 y) + 15 \cdot - 9$ .

$15 (y^{2} + 5 y - 9) = 0$

Step 5

Divide each term in $15 (y^{2} + 5 y - 9) = 0$ by $15$ and simplify.

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Step 5.1

Divide each term in $15 (y^{2} + 5 y - 9) = 0$ by $15$ .

$\frac{15 (y^{2} + 5 y - 9)}{15} = \frac{0}{15}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $15$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{15 (y^{2} + 5 y - 9)}{15} = \frac{0}{15}$

Step 5.2.1.2

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

$y^{2} + 5 y - 9 = \frac{0}{15}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $15$ .

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

Step 6

Use the quadratic formula to find the solutions.

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

Step 7

Substitute the values $a = 1$ , $b = 5$ , and $c = - 9$ into the quadratic formula and solve for $y$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (1 \cdot - 9)}}{2 \cdot 1}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $5$ to the power of $2$ .

$y = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 9}}{2 \cdot 1}$

Step 8.1.2

Multiply $- 4 \cdot 1 \cdot - 9$ .

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Step 8.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 9}}{2 \cdot 1}$

Step 8.1.2.2

Multiply $- 4$ by $- 9$ .

$y = \frac{- 5 \pm \sqrt{25 + 36}}{2 \cdot 1}$

Step 8.1.3

Add $25$ and $36$ .

$y = \frac{- 5 \pm \sqrt{61}}{2 \cdot 1}$

Step 8.2

Multiply $2$ by $1$ .

$y = \frac{- 5 \pm \sqrt{61}}{2}$

Step 9

The final answer is the combination of both solutions.

$y = - \frac{5 - \sqrt{61}}{2}, - \frac{5 + \sqrt{61}}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{5 - \sqrt{61}}{2}, - \frac{5 + \sqrt{61}}{2}$

Decimal Form:

$y = 1.40512483 \dots, - 6.40512483 \dots$