Solve 0=y^2-8y-371

$$0 = y ^ { 2 } - 8 y - 371$$

Solve for y

$$y=3\sqrt{43}+4\approx 23.672315573$$

$$y=4-3\sqrt{43}\approx -15.672315573$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$y^{2}-8y-371=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -8 for b, and -371 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-371\right)}}{2}$$

Square -8.

$$y=\frac{-\left(-8\right)±\sqrt{64-4\left(-371\right)}}{2}$$

Multiply -4 times -371.

$$y=\frac{-\left(-8\right)±\sqrt{64+1484}}{2}$$

Add 64 to 1484.

$$y=\frac{-\left(-8\right)±\sqrt{1548}}{2}$$

Take the square root of 1548.

$$y=\frac{-\left(-8\right)±6\sqrt{43}}{2}$$

The opposite of -8 is 8.

$$y=\frac{8±6\sqrt{43}}{2}$$

Now solve the equation y=\frac{8±6\sqrt{43}}{2} when ± is plus. Add 8 to 6\sqrt{43}.

$$y=\frac{6\sqrt{43}+8}{2}$$

Divide 8+6\sqrt{43} by 2.

$$y=3\sqrt{43}+4$$

Now solve the equation y=\frac{8±6\sqrt{43}}{2} when ± is minus. Subtract 6\sqrt{43} from 8.

$$y=\frac{8-6\sqrt{43}}{2}$$

Divide 8-6\sqrt{43} by 2.

$$y=4-3\sqrt{43}$$

The equation is now solved.

$$y=3\sqrt{43}+4 y=4-3\sqrt{43}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$y^{2}-8y-371=0$$

Add 371 to both sides. Anything plus zero gives itself.

$$y^{2}-8y=371$$

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.

$$y^{2}-8y+\left(-4\right)^{2}=371+\left(-4\right)^{2}$$

Square -4.

$$y^{2}-8y+16=371+16$$

Add 371 to 16.

$$y^{2}-8y+16=387$$

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}.

$$\left(y-4\right)^{2}=387$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-4\right)^{2}}=\sqrt{387}$$

Simplify.

$$y-4=3\sqrt{43} y-4=-3\sqrt{43}$$

Add 4 to both sides of the equation.

$$y=3\sqrt{43}+4 y=4-3\sqrt{43}$$