Solve y^2-2y-y+32=0

$$y ^ { 2 } - 2 y - y + 32 = 0$$

Solve for y

$$y=\frac{3+\sqrt{119}i}{2}\approx 1.5+5.454356057i$$

$$y=\frac{-\sqrt{119}i+3}{2}\approx 1.5-5.454356057i$$

Steps Using the Quadratic Formula

Combine -2y and -y to get -3y.

$$y^{2}-3y+32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and 32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 32}}{2}$$

Square -3.

$$y=\frac{-\left(-3\right)±\sqrt{9-4\times 32}}{2}$$

Multiply -4 times 32.

$$y=\frac{-\left(-3\right)±\sqrt{9-128}}{2}$$

Add 9 to -128.

$$y=\frac{-\left(-3\right)±\sqrt{-119}}{2}$$

Take the square root of -119.

$$y=\frac{-\left(-3\right)±\sqrt{119}i}{2}$$

The opposite of -3 is 3.

$$y=\frac{3±\sqrt{119}i}{2}$$

Now solve the equation y=\frac{3±\sqrt{119}i}{2} when ± is plus. Add 3 to i\sqrt{119}.

$$y=\frac{3+\sqrt{119}i}{2}$$

Now solve the equation y=\frac{3±\sqrt{119}i}{2} when ± is minus. Subtract i\sqrt{119} from 3.

$$y=\frac{-\sqrt{119}i+3}{2}$$

The equation is now solved.

$$y=\frac{3+\sqrt{119}i}{2} y=\frac{-\sqrt{119}i+3}{2}$$

Steps for Completing the Square

Combine -2y and -y to get -3y.

$$y^{2}-3y+32=0$$

Subtract 32 from both sides. Anything subtracted from zero gives its negation.

$$y^{2}-3y=-32$$

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

$$y^{2}-3y+\left(-\frac{3}{2}\right)^{2}=-32+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-3y+\frac{9}{4}=-32+\frac{9}{4}$$

Add -32 to \frac{9}{4}.

$$y^{2}-3y+\frac{9}{4}=-\frac{119}{4}$$

Factor y^{2}-3y+\frac{9}{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}.

$$\left(y-\frac{3}{2}\right)^{2}=-\frac{119}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{3}{2}\right)^{2}}=\sqrt{-\frac{119}{4}}$$

Simplify.

$$y-\frac{3}{2}=\frac{\sqrt{119}i}{2} y-\frac{3}{2}=-\frac{\sqrt{119}i}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$y=\frac{3+\sqrt{119}i}{2} y=\frac{-\sqrt{119}i+3}{2}$$