Finite Math Examples

${(3 y - 3)}^{2} + 6 (3 y - 3) - 72 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Let $u = 3 y - 3$ . Substitute $u$ for all occurrences of $3 y - 3$ .

$u^{2} + 6 u - 72 = 0$

Step 1.2

Factor $u^{2} + 6 u - 72$ 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 $- 72$ and whose sum is $6$ .

$- 6, 12$

Step 1.2.2

Write the factored form using these integers.

$(u - 6) (u + 12) = 0$

Step 1.3

Replace all occurrences of $u$ with $3 y - 3$ .

$(3 y - 3 - 6) (3 y - 3 + 12) = 0$

Step 2

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

$3 y - 3 - 6 = 0$

$3 y - 3 + 12 = 0$

Step 3

Set $3 y - 3 - 6$ equal to $0$ and solve for $y$ .

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Step 3.1

Set $3 y - 3 - 6$ equal to $0$ .

$3 y - 3 - 6 = 0$

Step 3.2

Solve $3 y - 3 - 6 = 0$ for $y$ .

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Step 3.2.1

Subtract $6$ from $- 3$ .

$3 y - 9 = 0$

Step 3.2.2

Add $9$ to both sides of the equation.

$3 y = 9$

Step 3.2.3

Divide each term in $3 y = 9$ by $3$ and simplify.

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Step 3.2.3.1

Divide each term in $3 y = 9$ by $3$ .

$\frac{3 y}{3} = \frac{9}{3}$

Step 3.2.3.2

Simplify the left side.

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Step 3.2.3.2.1

Cancel the common factor of $3$ .

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Step 3.2.3.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{9}{3}$

Step 3.2.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{9}{3}$

Step 3.2.3.3

Simplify the right side.

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Step 3.2.3.3.1

Divide $9$ by $3$ .

$y = 3$

Step 4

Set $3 y - 3 + 12$ equal to $0$ and solve for $y$ .

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Step 4.1

Set $3 y - 3 + 12$ equal to $0$ .

$3 y - 3 + 12 = 0$

Step 4.2

Solve $3 y - 3 + 12 = 0$ for $y$ .

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Step 4.2.1

Add $- 3$ and $12$ .

$3 y + 9 = 0$

Step 4.2.2

Subtract $9$ from both sides of the equation.

$3 y = - 9$

Step 4.2.3

Divide each term in $3 y = - 9$ by $3$ and simplify.

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Step 4.2.3.1

Divide each term in $3 y = - 9$ by $3$ .

$\frac{3 y}{3} = \frac{- 9}{3}$

Step 4.2.3.2

Simplify the left side.

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Step 4.2.3.2.1

Cancel the common factor of $3$ .

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Step 4.2.3.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{- 9}{3}$

Step 4.2.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 9}{3}$

Step 4.2.3.3

Simplify the right side.

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Step 4.2.3.3.1

Divide $- 9$ by $3$ .

$y = - 3$

Step 5

The final solution is all the values that make $(3 y - 3 - 6) (3 y - 3 + 12) = 0$ true.

$y = 3, - 3$