Precalculus Examples

$\frac{4 y^{2} - 8 y + 12}{y^{2} + y - 6} = 3$

Step 1

Factor each term.

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Step 1.1

Factor $4$ out of $4 y^{2} - 8 y + 12$ .

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Step 1.1.1

Factor $4$ out of $4 y^{2}$ .

$\frac{4 (y^{2}) - 8 y + 12}{y^{2} + y - 6} = 3$

Step 1.1.2

Factor $4$ out of $- 8 y$ .

$\frac{4 (y^{2}) + 4 (- 2 y) + 12}{y^{2} + y - 6} = 3$

Step 1.1.3

Factor $4$ out of $12$ .

$\frac{4 y^{2} + 4 (- 2 y) + 4 \cdot 3}{y^{2} + y - 6} = 3$

Step 1.1.4

Factor $4$ out of $4 y^{2} + 4 (- 2 y)$ .

$\frac{4 (y^{2} - 2 y) + 4 \cdot 3}{y^{2} + y - 6} = 3$

Step 1.1.5

Factor $4$ out of $4 (y^{2} - 2 y) + 4 \cdot 3$ .

$\frac{4 (y^{2} - 2 y + 3)}{y^{2} + y - 6} = 3$

Step 1.2

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

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Step 1.2.1

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 1.2.2

Write the factored form using these integers.

$\frac{4 (y^{2} - 2 y + 3)}{(y - 2) (y + 3)} = 3$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$(y - 2) (y + 3), 1$

Step 2.2

The LCM of one and any expression is the expression.

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

Step 3

Multiply each term in $\frac{4 (y^{2} - 2 y + 3)}{(y - 2) (y + 3)} = 3$ by $(y - 2) (y + 3)$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{4 (y^{2} - 2 y + 3)}{(y - 2) (y + 3)} = 3$ by $(y - 2) (y + 3)$ .

$\frac{4 (y^{2} - 2 y + 3)}{(y - 2) (y + 3)} ((y - 2) (y + 3)) = 3 ((y - 2) (y + 3))$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $(y - 2) (y + 3)$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{4 (y^{2} - 2 y + 3)}{(y - 2) (y + 3)} ((y - 2) (y + 3)) = 3 ((y - 2) (y + 3))$

Step 3.2.1.2

Rewrite the expression.

$4 (y^{2} - 2 y + 3) = 3 ((y - 2) (y + 3))$

Step 3.2.2

Apply the distributive property.

$4 y^{2} + 4 (- 2 y) + 4 \cdot 3 = 3 ((y - 2) (y + 3))$

Step 3.2.3

Simplify.

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Step 3.2.3.1

Multiply $- 2$ by $4$ .

$4 y^{2} - 8 y + 4 \cdot 3 = 3 ((y - 2) (y + 3))$

Step 3.2.3.2

Multiply $4$ by $3$ .

$4 y^{2} - 8 y + 12 = 3 ((y - 2) (y + 3))$

Step 3.3

Simplify the right side.

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Step 3.3.1

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

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Step 3.3.1.1

Apply the distributive property.

$4 y^{2} - 8 y + 12 = 3 (y (y + 3) - 2 (y + 3))$

Step 3.3.1.2

Apply the distributive property.

$4 y^{2} - 8 y + 12 = 3 (y \cdot y + y \cdot 3 - 2 (y + 3))$

Step 3.3.1.3

Apply the distributive property.

$4 y^{2} - 8 y + 12 = 3 (y \cdot y + y \cdot 3 - 2 y - 2 \cdot 3)$

Step 3.3.2

Simplify and combine like terms.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Multiply $y$ by $y$ .

$4 y^{2} - 8 y + 12 = 3 (y^{2} + y \cdot 3 - 2 y - 2 \cdot 3)$

Step 3.3.2.1.2

Move $3$ to the left of $y$ .

$4 y^{2} - 8 y + 12 = 3 (y^{2} + 3 \cdot y - 2 y - 2 \cdot 3)$

Step 3.3.2.1.3

Multiply $- 2$ by $3$ .

$4 y^{2} - 8 y + 12 = 3 (y^{2} + 3 y - 2 y - 6)$

Step 3.3.2.2

Subtract $2 y$ from $3 y$ .

$4 y^{2} - 8 y + 12 = 3 (y^{2} + y - 6)$

Step 3.3.3

Apply the distributive property.

$4 y^{2} - 8 y + 12 = 3 y^{2} + 3 y + 3 \cdot - 6$

Step 3.3.4

Multiply $3$ by $- 6$ .

$4 y^{2} - 8 y + 12 = 3 y^{2} + 3 y - 18$

Step 4

Solve the equation.

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Step 4.1

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

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Step 4.1.1

Subtract $3 y^{2}$ from both sides of the equation.

$4 y^{2} - 8 y + 12 - 3 y^{2} = 3 y - 18$

Step 4.1.2

Subtract $3 y$ from both sides of the equation.

$4 y^{2} - 8 y + 12 - 3 y^{2} - 3 y = - 18$

Step 4.1.3

Subtract $3 y^{2}$ from $4 y^{2}$ .

$y^{2} - 8 y + 12 - 3 y = - 18$

Step 4.1.4

Subtract $3 y$ from $- 8 y$ .

$y^{2} - 11 y + 12 = - 18$

Step 4.2

Add $18$ to both sides of the equation.

$y^{2} - 11 y + 12 + 18 = 0$

Step 4.3

Add $12$ and $18$ .

$y^{2} - 11 y + 30 = 0$

Step 4.4

Factor $y^{2} - 11 y + 30$ using the AC method.

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Step 4.4.1

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 $30$ and whose sum is $- 11$ .

$- 6, - 5$

Step 4.4.2

Write the factored form using these integers.

$(y - 6) (y - 5) = 0$

Step 4.5

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

$y - 6 = 0$

$y - 5 = 0$

Step 4.6

Set $y - 6$ equal to $0$ and solve for $y$ .

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Step 4.6.1

Set $y - 6$ equal to $0$ .

$y - 6 = 0$

Step 4.6.2

Add $6$ to both sides of the equation.

$y = 6$

Step 4.7

Set $y - 5$ equal to $0$ and solve for $y$ .

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Step 4.7.1

Set $y - 5$ equal to $0$ .

$y - 5 = 0$

Step 4.7.2

Add $5$ to both sides of the equation.

$y = 5$

Step 4.8

The final solution is all the values that make $(y - 6) (y - 5) = 0$ true.

$y = 6, 5$