Solve 3div(y-7)-(2y+9)=13

$$3 \div (y-7)-(2y+9)=13$$

Solve for y

$$y=\frac{\sqrt{330}}{2}-2\approx 7.082951062$$

$$y=-\frac{\sqrt{330}}{2}-2\approx -11.082951062$$

Steps Using the Quadratic Formula

Variable y cannot be equal to 7 since division by zero is not defined. Multiply both sides of the equation by y-7.

$$3-\left(2y+9\right)\left(y-7\right)=13\left(y-7\right)$$

Use the distributive property to multiply -1 by 2y+9.

$$3+\left(-2y-9\right)\left(y-7\right)=13\left(y-7\right)$$

Use the distributive property to multiply -2y-9 by y-7 and combine like terms.

$$3-2y^{2}+5y+63=13\left(y-7\right)$$

Add 3 and 63 to get 66.

$$66-2y^{2}+5y=13\left(y-7\right)$$

Use the distributive property to multiply 13 by y-7.

$$66-2y^{2}+5y=13y-91$$

Subtract 13y from both sides.

$$66-2y^{2}+5y-13y=-91$$

Combine 5y and -13y to get -8y.

$$66-2y^{2}-8y=-91$$

Add 91 to both sides.

$$66-2y^{2}-8y+91=0$$

Add 66 and 91 to get 157.

$$157-2y^{2}-8y=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-2y^{2}-8y+157=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -8 for b, and 157 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-2\right)\times 157}}{2\left(-2\right)}$$

Square -8.

$$y=\frac{-\left(-8\right)±\sqrt{64-4\left(-2\right)\times 157}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$y=\frac{-\left(-8\right)±\sqrt{64+8\times 157}}{2\left(-2\right)}$$

Multiply 8 times 157.

$$y=\frac{-\left(-8\right)±\sqrt{64+1256}}{2\left(-2\right)}$$

Add 64 to 1256.

$$y=\frac{-\left(-8\right)±\sqrt{1320}}{2\left(-2\right)}$$

Take the square root of 1320.

$$y=\frac{-\left(-8\right)±2\sqrt{330}}{2\left(-2\right)}$$

The opposite of -8 is 8.

$$y=\frac{8±2\sqrt{330}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$y=\frac{8±2\sqrt{330}}{-4}$$

Now solve the equation y=\frac{8±2\sqrt{330}}{-4} when ± is plus. Add 8 to 2\sqrt{330}.

$$y=\frac{2\sqrt{330}+8}{-4}$$

Divide 8+2\sqrt{330} by -4.

$$y=-\frac{\sqrt{330}}{2}-2$$

Now solve the equation y=\frac{8±2\sqrt{330}}{-4} when ± is minus. Subtract 2\sqrt{330} from 8.

$$y=\frac{8-2\sqrt{330}}{-4}$$

Divide 8-2\sqrt{330} by -4.

$$y=\frac{\sqrt{330}}{2}-2$$

The equation is now solved.

$$y=-\frac{\sqrt{330}}{2}-2 y=\frac{\sqrt{330}}{2}-2$$

Steps for Completing the Square

Variable y cannot be equal to 7 since division by zero is not defined. Multiply both sides of the equation by y-7.

$$3-\left(2y+9\right)\left(y-7\right)=13\left(y-7\right)$$

Use the distributive property to multiply -1 by 2y+9.

$$3+\left(-2y-9\right)\left(y-7\right)=13\left(y-7\right)$$

Use the distributive property to multiply -2y-9 by y-7 and combine like terms.

$$3-2y^{2}+5y+63=13\left(y-7\right)$$

Add 3 and 63 to get 66.

$$66-2y^{2}+5y=13\left(y-7\right)$$

Use the distributive property to multiply 13 by y-7.

$$66-2y^{2}+5y=13y-91$$

Subtract 13y from both sides.

$$66-2y^{2}+5y-13y=-91$$

Combine 5y and -13y to get -8y.

$$66-2y^{2}-8y=-91$$

Subtract 66 from both sides.

$$-2y^{2}-8y=-91-66$$

Subtract 66 from -91 to get -157.

$$-2y^{2}-8y=-157$$

Divide both sides by -2.

$$\frac{-2y^{2}-8y}{-2}=-\frac{157}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$y^{2}+\left(-\frac{8}{-2}\right)y=-\frac{157}{-2}$$

Divide -8 by -2.

$$y^{2}+4y=-\frac{157}{-2}$$

Divide -157 by -2.

$$y^{2}+4y=\frac{157}{2}$$

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+4y+2^{2}=\frac{157}{2}+2^{2}$$

Square 2.

$$y^{2}+4y+4=\frac{157}{2}+4$$

Add \frac{157}{2} to 4.

$$y^{2}+4y+4=\frac{165}{2}$$

Factor y^{2}+4y+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+2\right)^{2}=\frac{165}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+2\right)^{2}}=\sqrt{\frac{165}{2}}$$

Simplify.

$$y+2=\frac{\sqrt{330}}{2} y+2=-\frac{\sqrt{330}}{2}$$

Subtract 2 from both sides of the equation.

$$y=\frac{\sqrt{330}}{2}-2 y=-\frac{\sqrt{330}}{2}-2$$