Solve 8(-0.5y+1)3y=31

$$8 ( - 0.5 y + 1 ) 3 y = 31$$

Solve for y

$$y=\frac{i\sqrt{57}}{6}+1\approx 1+1.258305739i$$

$$y=-\frac{i\sqrt{57}}{6}+1\approx 1-1.258305739i$$

Steps Using the Quadratic Formula

Multiply 8 and 3 to get 24.

$$24\left(-0.5y+1\right)y=31$$

Use the distributive property to multiply 24 by -0.5y+1.

$$\left(-12y+24\right)y=31$$

Use the distributive property to multiply -12y+24 by y.

$$-12y^{2}+24y=31$$

Subtract 31 from both sides.

$$-12y^{2}+24y-31=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -12 for a, 24 for b, and -31 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-24±\sqrt{24^{2}-4\left(-12\right)\left(-31\right)}}{2\left(-12\right)}$$

Square 24.

$$y=\frac{-24±\sqrt{576-4\left(-12\right)\left(-31\right)}}{2\left(-12\right)}$$

Multiply -4 times -12.

$$y=\frac{-24±\sqrt{576+48\left(-31\right)}}{2\left(-12\right)}$$

Multiply 48 times -31.

$$y=\frac{-24±\sqrt{576-1488}}{2\left(-12\right)}$$

Add 576 to -1488.

$$y=\frac{-24±\sqrt{-912}}{2\left(-12\right)}$$

Take the square root of -912.

$$y=\frac{-24±4\sqrt{57}i}{2\left(-12\right)}$$

Multiply 2 times -12.

$$y=\frac{-24±4\sqrt{57}i}{-24}$$

Now solve the equation y=\frac{-24±4\sqrt{57}i}{-24} when ± is plus. Add -24 to 4i\sqrt{57}.

$$y=\frac{-24+4\sqrt{57}i}{-24}$$

Divide -24+4i\sqrt{57} by -24.

$$y=-\frac{\sqrt{57}i}{6}+1$$

Now solve the equation y=\frac{-24±4\sqrt{57}i}{-24} when ± is minus. Subtract 4i\sqrt{57} from -24.

$$y=\frac{-4\sqrt{57}i-24}{-24}$$

Divide -24-4i\sqrt{57} by -24.

$$y=\frac{\sqrt{57}i}{6}+1$$

The equation is now solved.

$$y=-\frac{\sqrt{57}i}{6}+1 y=\frac{\sqrt{57}i}{6}+1$$

Steps for Completing the Square

Multiply 8 and 3 to get 24.

$$24\left(-0.5y+1\right)y=31$$

Use the distributive property to multiply 24 by -0.5y+1.

$$\left(-12y+24\right)y=31$$

Use the distributive property to multiply -12y+24 by y.

$$-12y^{2}+24y=31$$

Divide both sides by -12.

$$\frac{-12y^{2}+24y}{-12}=\frac{31}{-12}$$

Dividing by -12 undoes the multiplication by -12.

$$y^{2}+\frac{24}{-12}y=\frac{31}{-12}$$

Divide 24 by -12.

$$y^{2}-2y=\frac{31}{-12}$$

Divide 31 by -12.

$$y^{2}-2y=-\frac{31}{12}$$

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

$$y^{2}-2y+1=-\frac{31}{12}+1$$

Add -\frac{31}{12} to 1.

$$y^{2}-2y+1=-\frac{19}{12}$$

Factor y^{2}-2y+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-1\right)^{2}=-\frac{19}{12}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-1\right)^{2}}=\sqrt{-\frac{19}{12}}$$

Simplify.

$$y-1=\frac{\sqrt{57}i}{6} y-1=-\frac{\sqrt{57}i}{6}$$

Add 1 to both sides of the equation.

$$y=\frac{\sqrt{57}i}{6}+1 y=-\frac{\sqrt{57}i}{6}+1$$