Solve for y (3y-7)^2+3(3y-7)-18=0

${(3 y - 7)}^{2} + 3 (3 y - 7) - 18 = 0$

Step 1

Factor the left side of the equation.

Let $u = 3 y - 7$ . Substitute $u$ for all occurrences of $3 y - 7$ .

$u^{2} + 3 u - 18 = 0$

Factor $u^{2} + 3 u - 18$ using the AC method.

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 $- 18$ and whose sum is $3$ .

$- 3, 6$

Write the factored form using these integers.

$(u - 3) (u + 6) = 0$

Replace all occurrences of $u$ with $3 y - 7$ .

$(3 y - 7 - 3) (3 y - 7 + 6) = 0$

Step 2

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

$3 y - 7 - 3 = 0$

$3 y - 7 + 6 = 0$

Step 3

Set $3 y - 7 - 3$ equal to $0$ and solve for $y$ .

Set $3 y - 7 - 3$ equal to $0$ .

$3 y - 7 - 3 = 0$

Solve $3 y - 7 - 3 = 0$ for $y$ .

Subtract $3$ from $- 7$ .

$3 y - 10 = 0$

Add $10$ to both sides of the equation.

$3 y = 10$

Divide each term in $3 y = 10$ by $3$ and simplify.

Divide each term in $3 y = 10$ by $3$ .

$\frac{3 y}{3} = \frac{10}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 y}{3} = \frac{10}{3}$

Divide $y$ by $1$ .

$y = \frac{10}{3}$

Step 4

Set $3 y - 7 + 6$ equal to $0$ and solve for $y$ .

Set $3 y - 7 + 6$ equal to $0$ .

$3 y - 7 + 6 = 0$

Solve $3 y - 7 + 6 = 0$ for $y$ .

Add $- 7$ and $6$ .

$3 y - 1 = 0$

Add $1$ to both sides of the equation.

$3 y = 1$

Divide each term in $3 y = 1$ by $3$ and simplify.

Divide each term in $3 y = 1$ by $3$ .

$\frac{3 y}{3} = \frac{1}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 y}{3} = \frac{1}{3}$

Divide $y$ by $1$ .

$y = \frac{1}{3}$

Step 5

The final solution is all the values that make $(3 y - 7 - 3) (3 y - 7 + 6) = 0$ true.

$y = \frac{10}{3}, \frac{1}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$y = \frac{10}{3}, \frac{1}{3}$

Decimal Form:

$y = 3.\overline{3}, 0.\overline{3}$

Mixed Number Form:

$y = 3 \frac{1}{3}, \frac{1}{3}$

Solve for y (3y-7)^2+3(3y-7)-18=0

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