Basic Math Examples

$y (3 y + 2) = 9$

Step 1

Simplify $y (3 y + 2)$ .

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Step 1.1

Simplify by multiplying through.

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Step 1.1.1

Apply the distributive property.

$y (3 y) + y \cdot 2 = 9$

Step 1.1.2

Reorder.

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Step 1.1.2.1

Rewrite using the commutative property of multiplication.

$3 y \cdot y + y \cdot 2 = 9$

Step 1.1.2.2

Move $2$ to the left of $y$ .

$3 y \cdot y + 2 \cdot y = 9$

Step 1.2

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

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Step 1.2.1

Move $y$ .

$3 (y \cdot y) + 2 \cdot y = 9$

Step 1.2.2

Multiply $y$ by $y$ .

$3 y^{2} + 2 \cdot y = 9$

$3 y^{2} + 2 y = 9$

Step 2

Subtract $9$ from both sides of the equation.

$3 y^{2} + 2 y - 9 = 0$

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 $a = 3$ , $b = 2$ , and $c = - 9$ into the quadratic formula and solve for $y$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (3 \cdot - 9)}}{2 \cdot 3}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

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

Step 5.1.2

Multiply $- 4 \cdot 3 \cdot - 9$ .

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Step 5.1.2.1

Multiply $- 4$ by $3$ .

$y = \frac{- 2 \pm \sqrt{4 - 12 \cdot - 9}}{2 \cdot 3}$

Step 5.1.2.2

Multiply $- 12$ by $- 9$ .

$y = \frac{- 2 \pm \sqrt{4 + 108}}{2 \cdot 3}$

Step 5.1.3

Add $4$ and $108$ .

$y = \frac{- 2 \pm \sqrt{112}}{2 \cdot 3}$

Step 5.1.4

Rewrite $112$ as $4^{2} \cdot 7$ .

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Step 5.1.4.1

Factor $16$ out of $112$ .

$y = \frac{- 2 \pm \sqrt{16 (7)}}{2 \cdot 3}$

Step 5.1.4.2

Rewrite $16$ as $4^{2}$ .

$y = \frac{- 2 \pm \sqrt{4^{2} \cdot 7}}{2 \cdot 3}$

Step 5.1.5

Pull terms out from under the radical.

$y = \frac{- 2 \pm 4 \sqrt{7}}{2 \cdot 3}$

Step 5.2

Multiply $2$ by $3$ .

$y = \frac{- 2 \pm 4 \sqrt{7}}{6}$

Step 5.3

Simplify $\frac{- 2 \pm 4 \sqrt{7}}{6}$ .

$y = \frac{- 1 \pm 2 \sqrt{7}}{3}$

Step 6

The final answer is the combination of both solutions.

$y = - \frac{1 - 2 \sqrt{7}}{3}, - \frac{1 + 2 \sqrt{7}}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{1 - 2 \sqrt{7}}{3}, - \frac{1 + 2 \sqrt{7}}{3}$

Decimal Form:

$y = 1.43050087 \dots, - 2.09716754 \dots$