Solve for y (2y-3)(y-2)=-12y+18

$(2 y - 3) (y - 2) = - 12 y + 18$

Step 1

Simplify $(2 y - 3) (y - 2)$ .

Rewrite.

$0 + 0 + (2 y - 3) (y - 2) = - 12 y + 18$

Simplify by adding zeros.

$(2 y - 3) (y - 2) = - 12 y + 18$

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

Apply the distributive property.

$2 y (y - 2) - 3 (y - 2) = - 12 y + 18$

Apply the distributive property.

$2 y \cdot y + 2 y \cdot - 2 - 3 (y - 2) = - 12 y + 18$

Apply the distributive property.

$2 y \cdot y + 2 y \cdot - 2 - 3 y - 3 \cdot - 2 = - 12 y + 18$

Simplify and combine like terms.

Simplify each term.

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

Move $y$ .

$2 (y \cdot y) + 2 y \cdot - 2 - 3 y - 3 \cdot - 2 = - 12 y + 18$

Multiply $y$ by $y$ .

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

Multiply $- 2$ by $2$ .

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

Multiply $- 3$ by $- 2$ .

$2 y^{2} - 4 y - 3 y + 6 = - 12 y + 18$

Subtract $3 y$ from $- 4 y$ .

$2 y^{2} - 7 y + 6 = - 12 y + 18$

Step 2

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

Add $12 y$ to both sides of the equation.

$2 y^{2} - 7 y + 6 + 12 y = 18$

Add $- 7 y$ and $12 y$ .

$2 y^{2} + 5 y + 6 = 18$

Step 3

Subtract $18$ from both sides of the equation.

$2 y^{2} + 5 y + 6 - 18 = 0$

Step 4

Subtract $18$ from $6$ .

$2 y^{2} + 5 y - 12 = 0$

Step 5

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 12 = - 24$ and whose sum is $b = 5$ .

Factor $5$ out of $5 y$ .

$2 y^{2} + 5 (y) - 12 = 0$

Rewrite $5$ as $- 3$ plus $8$

$2 y^{2} + (- 3 + 8) y - 12 = 0$

Apply the distributive property.

$2 y^{2} - 3 y + 8 y - 12 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(2 y^{2} - 3 y) + 8 y - 12 = 0$

Factor out the greatest common factor (GCF) from each group.

$y (2 y - 3) + 4 (2 y - 3) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 y - 3$ .

$(2 y - 3) (y + 4) = 0$

Step 6

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

$2 y - 3 = 0$

$y + 4 = 0$

Step 7

Set $2 y - 3$ equal to $0$ and solve for $y$ .

Set $2 y - 3$ equal to $0$ .

$2 y - 3 = 0$

Solve $2 y - 3 = 0$ for $y$ .

Add $3$ to both sides of the equation.

$2 y = 3$

Divide each term in $2 y = 3$ by $2$ and simplify.

Divide each term in $2 y = 3$ by $2$ .

$\frac{2 y}{2} = \frac{3}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y}{2} = \frac{3}{2}$

Divide $y$ by $1$ .

$y = \frac{3}{2}$

Step 8

Set $y + 4$ equal to $0$ and solve for $y$ .

Set $y + 4$ equal to $0$ .

$y + 4 = 0$

Subtract $4$ from both sides of the equation.

$y = - 4$

Step 9

The final solution is all the values that make $(2 y - 3) (y + 4) = 0$ true.

$y = \frac{3}{2}, - 4$

Step 10

The result can be shown in multiple forms.

Exact Form:

$y = \frac{3}{2}, - 4$

Decimal Form:

$y = 1.5, - 4$

Mixed Number Form:

$y = 1 \frac{1}{2}, - 4$

Solve for y (2y-3)(y-2)=-12y+18

Related Questions

Support LCMGCF.com by disabling your adblocker.