Solve y^2-2y-8=-y+6

$$y ^ { 2 } - 2 y - 8 = - y + 6$$

Solve for y

$$y = \frac{\sqrt{57} + 1}{2} \approx 4.274917218$$

$$y=\frac{1-\sqrt{57}}{2}\approx -3.274917218$$

Steps Using the Quadratic Formula

Add y to both sides.

$$y^{2}-2y-8+y=6$$

Combine -2y and y to get -y.

$$y^{2}-y-8=6$$

Subtract 6 from both sides.

$$y^{2}-y-8-6=0$$

Subtract 6 from -8 to get -14.

$$y^{2}-y-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-1\right)±\sqrt{1-4\left(-14\right)}}{2}$$

Multiply -4 times -14.

$$y=\frac{-\left(-1\right)±\sqrt{1+56}}{2}$$

Add 1 to 56.

$$y=\frac{-\left(-1\right)±\sqrt{57}}{2}$$

The opposite of -1 is 1.

$$y=\frac{1±\sqrt{57}}{2}$$

Now solve the equation y=\frac{1±\sqrt{57}}{2} when ± is plus. Add 1 to \sqrt{57}.

$$y=\frac{\sqrt{57}+1}{2}$$

Now solve the equation y=\frac{1±\sqrt{57}}{2} when ± is minus. Subtract \sqrt{57} from 1.

$$y=\frac{1-\sqrt{57}}{2}$$

The equation is now solved.

$$y=\frac{\sqrt{57}+1}{2} y=\frac{1-\sqrt{57}}{2}$$

Steps for Completing the Square

Add y to both sides.

$$y^{2}-2y-8+y=6$$

Combine -2y and y to get -y.

$$y^{2}-y-8=6$$

Add 8 to both sides.

$$y^{2}-y=6+8$$

Add 6 and 8 to get 14.

$$y^{2}-y=14$$

Divide -1, the coefficient of the x term, by 2 to get -\frac{1}{2}. Then add the square of -\frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-y+\left(-\frac{1}{2}\right)^{2}=14+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-y+\frac{1}{4}=14+\frac{1}{4}$$

Add 14 to \frac{1}{4}.

$$y^{2}-y+\frac{1}{4}=\frac{57}{4}$$

Factor y^{2}-y+\frac{1}{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}.

$$\left(y-\frac{1}{2}\right)^{2}=\frac{57}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{1}{2}\right)^{2}}=\sqrt{\frac{57}{4}}$$

Simplify.

$$y-\frac{1}{2}=\frac{\sqrt{57}}{2} y-\frac{1}{2}=-\frac{\sqrt{57}}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$y=\frac{\sqrt{57}+1}{2} y=\frac{1-\sqrt{57}}{2}$$