Pre-Algebra Examples

$2 y^{2} - (y + 2) (y - 3) = 12$

Step 1

Simplify $2 y^{2} - (y + 2) (y - 3)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$2 y^{2} + (- y - 1 \cdot 2) (y - 3) = 12$

Step 1.1.2

Multiply $- 1$ by $2$ .

$2 y^{2} + (- y - 2) (y - 3) = 12$

Step 1.1.3

Expand $(- y - 2) (y - 3)$ using the FOIL Method.

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Step 1.1.3.1

Apply the distributive property.

$2 y^{2} - y (y - 3) - 2 (y - 3) = 12$

Step 1.1.3.2

Apply the distributive property.

$2 y^{2} - y \cdot y - y \cdot - 3 - 2 (y - 3) = 12$

Step 1.1.3.3

Apply the distributive property.

$2 y^{2} - y \cdot y - y \cdot - 3 - 2 y - 2 \cdot - 3 = 12$

Step 1.1.4

Simplify and combine like terms.

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Step 1.1.4.1

Simplify each term.

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Step 1.1.4.1.1

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

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Step 1.1.4.1.1.1

Move $y$ .

$2 y^{2} - (y \cdot y) - y \cdot - 3 - 2 y - 2 \cdot - 3 = 12$

Step 1.1.4.1.1.2

Multiply $y$ by $y$ .

$2 y^{2} - y^{2} - y \cdot - 3 - 2 y - 2 \cdot - 3 = 12$

Step 1.1.4.1.2

Multiply $- 3$ by $- 1$ .

$2 y^{2} - y^{2} + 3 y - 2 y - 2 \cdot - 3 = 12$

Step 1.1.4.1.3

Multiply $- 2$ by $- 3$ .

$2 y^{2} - y^{2} + 3 y - 2 y + 6 = 12$

Step 1.1.4.2

Subtract $2 y$ from $3 y$ .

$2 y^{2} - y^{2} + y + 6 = 12$

Step 1.2

Subtract $y^{2}$ from $2 y^{2}$ .

$y^{2} + y + 6 = 12$

Step 2

Subtract $12$ from both sides of the equation.

$y^{2} + y + 6 - 12 = 0$

Step 3

Subtract $12$ from $6$ .

$y^{2} + y - 6 = 0$

Step 4

Factor $y^{2} + y - 6$ using the AC method.

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Step 4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 4.2

Write the factored form using these integers.

$(y - 2) (y + 3) = 0$

Step 5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y - 2 = 0$

$y + 3 = 0$

Step 6

Set $y - 2$ equal to $0$ and solve for $y$ .

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Step 6.1

Set $y - 2$ equal to $0$ .

$y - 2 = 0$

Step 6.2

Add $2$ to both sides of the equation.

$y = 2$

Step 7

Set $y + 3$ equal to $0$ and solve for $y$ .

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Step 7.1

Set $y + 3$ equal to $0$ .

$y + 3 = 0$

Step 7.2

Subtract $3$ from both sides of the equation.

$y = - 3$

Step 8

The final solution is all the values that make $(y - 2) (y + 3) = 0$ true.

$y = 2, - 3$