Solve y^2+8y+9/2=19

$$\frac { y ^ { 2 } + 8 y + 9 } { 2 } = 19$$

Solve for y

$$y=3\sqrt{5}-4\approx 2.708203932$$

$$y=-3\sqrt{5}-4\approx -10.708203932$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$y^{2}+8y+9=19\times 2$$

Multiply 19 and 2 to get 38.

$$y^{2}+8y+9=38$$

Subtract 38 from both sides.

$$y^{2}+8y+9-38=0$$

Subtract 38 from 9 to get -29.

$$y^{2}+8y-29=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and -29 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-8±\sqrt{8^{2}-4\left(-29\right)}}{2}$$

Square 8.

$$y=\frac{-8±\sqrt{64-4\left(-29\right)}}{2}$$

Multiply -4 times -29.

$$y=\frac{-8±\sqrt{64+116}}{2}$$

Add 64 to 116.

$$y=\frac{-8±\sqrt{180}}{2}$$

Take the square root of 180.

$$y=\frac{-8±6\sqrt{5}}{2}$$

Now solve the equation y=\frac{-8±6\sqrt{5}}{2} when ± is plus. Add -8 to 6\sqrt{5}.

$$y=\frac{6\sqrt{5}-8}{2}$$

Divide -8+6\sqrt{5} by 2.

$$y=3\sqrt{5}-4$$

Now solve the equation y=\frac{-8±6\sqrt{5}}{2} when ± is minus. Subtract 6\sqrt{5} from -8.

$$y=\frac{-6\sqrt{5}-8}{2}$$

Divide -8-6\sqrt{5} by 2.

$$y=-3\sqrt{5}-4$$

The equation is now solved.

$$y=3\sqrt{5}-4 y=-3\sqrt{5}-4$$

Steps for Completing the Square

Multiply both sides by 2.

$$y^{2}+8y+9=19\times 2$$

Multiply 19 and 2 to get 38.

$$y^{2}+8y+9=38$$

Subtract 9 from both sides.

$$y^{2}+8y=38-9$$

Subtract 9 from 38 to get 29.

$$y^{2}+8y=29$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+8y+4^{2}=29+4^{2}$$

Square 4.

$$y^{2}+8y+16=29+16$$

Add 29 to 16.

$$y^{2}+8y+16=45$$

Factor y^{2}+8y+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+4\right)^{2}=45$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+4\right)^{2}}=\sqrt{45}$$

Simplify.

$$y+4=3\sqrt{5} y+4=-3\sqrt{5}$$

Subtract 4 from both sides of the equation.

$$y=3\sqrt{5}-4 y=-3\sqrt{5}-4$$