Solve (2+y)(8-y)=9

$$( 2 + y ) ( 8 - y ) = 9$$

Solve for y

$$y=-1$$

$$y=7$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2+y by 8-y and combine like terms.

$$16+6y-y^{2}=9$$

Subtract 9 from both sides.

$$16+6y-y^{2}-9=0$$

Subtract 9 from 16 to get 7.

$$7+6y-y^{2}=0$$

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.

$$-y^{2}+6y+7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 6 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-6±\sqrt{6^{2}-4\left(-1\right)\times 7}}{2\left(-1\right)}$$

Square 6.

$$y=\frac{-6±\sqrt{36-4\left(-1\right)\times 7}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$y=\frac{-6±\sqrt{36+4\times 7}}{2\left(-1\right)}$$

Multiply 4 times 7.

$$y=\frac{-6±\sqrt{36+28}}{2\left(-1\right)}$$

Add 36 to 28.

$$y=\frac{-6±\sqrt{64}}{2\left(-1\right)}$$

Take the square root of 64.

$$y=\frac{-6±8}{2\left(-1\right)}$$

Multiply 2 times -1.

$$y=\frac{-6±8}{-2}$$

Now solve the equation y=\frac{-6±8}{-2} when ± is plus. Add -6 to 8.

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

Divide 2 by -2.

$$y=-1$$

Now solve the equation y=\frac{-6±8}{-2} when ± is minus. Subtract 8 from -6.

$$y=-\frac{14}{-2}$$

Divide -14 by -2.

$$y=7$$

The equation is now solved.

$$y=-1 y=7$$

Steps for Completing the Square

Use the distributive property to multiply 2+y by 8-y and combine like terms.

$$16+6y-y^{2}=9$$

Subtract 16 from both sides.

$$6y-y^{2}=9-16$$

Subtract 16 from 9 to get -7.

$$6y-y^{2}=-7$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

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

Divide both sides by -1.

$$\frac{-y^{2}+6y}{-1}=-\frac{7}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$y^{2}+\frac{6}{-1}y=-\frac{7}{-1}$$

Divide 6 by -1.

$$y^{2}-6y=-\frac{7}{-1}$$

Divide -7 by -1.

$$y^{2}-6y=7$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-6y+\left(-3\right)^{2}=7+\left(-3\right)^{2}$$

Square -3.

$$y^{2}-6y+9=7+9$$

Add 7 to 9.

$$y^{2}-6y+9=16$$

Factor y^{2}-6y+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-3\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-3\right)^{2}}=\sqrt{16}$$

Simplify.

$$y-3=4 y-3=-4$$

Add 3 to both sides of the equation.

$$y=7 y=-1$$