Solve 20=y(y+1)

$$20 = y ( y + 1 )$$

Solve for y

$$y=4$$

$$y=-5$$

Steps Using the Quadratic Formula

Use the distributive property to multiply y by y+1.

$$20=y^{2}+y$$

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

$$y^{2}+y=20$$

Subtract 20 from both sides.

$$y^{2}+y-20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and -20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-1±\sqrt{1^{2}-4\left(-20\right)}}{2}$$

Square 1.

$$y=\frac{-1±\sqrt{1-4\left(-20\right)}}{2}$$

Multiply -4 times -20.

$$y=\frac{-1±\sqrt{1+80}}{2}$$

Add 1 to 80.

$$y=\frac{-1±\sqrt{81}}{2}$$

Take the square root of 81.

$$y=\frac{-1±9}{2}$$

Now solve the equation y=\frac{-1±9}{2} when ± is plus. Add -1 to 9.

$$y=\frac{8}{2}$$

Divide 8 by 2.

$$y=4$$

Now solve the equation y=\frac{-1±9}{2} when ± is minus. Subtract 9 from -1.

$$y=-\frac{10}{2}$$

Divide -10 by 2.

$$y=-5$$

The equation is now solved.

$$y=4 y=-5$$

Steps for Completing the Square

Use the distributive property to multiply y by y+1.

$$20=y^{2}+y$$

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

$$y^{2}+y=20$$

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.

$$y^{2}+y+\left(\frac{1}{2}\right)^{2}=20+\left(\frac{1}{2}\right)^{2}$$

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

$$y^{2}+y+\frac{1}{4}=20+\frac{1}{4}$$

Add 20 to \frac{1}{4}.

$$y^{2}+y+\frac{1}{4}=\frac{81}{4}$$

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}.

$$\left(y+\frac{1}{2}\right)^{2}=\frac{81}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{1}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Simplify.

$$y+\frac{1}{2}=\frac{9}{2} y+\frac{1}{2}=-\frac{9}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$y=4 y=-5$$