Solve (y+2)(y-3)=1

$$( y + 2 ) ( y - 3 ) = 1$$

Solve for y

$$y = \frac{\sqrt{29} + 1}{2} \approx 3.192582404$$

$$y=\frac{1-\sqrt{29}}{2}\approx -2.192582404$$

Steps Using the Quadratic Formula

Use the distributive property to multiply y+2 by y-3 and combine like terms.

$$y^{2}-y-6=1$$

Subtract 1 from both sides.

$$y^{2}-y-6-1=0$$

Subtract 1 from -6 to get -7.

$$y^{2}-y-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-1\right)±\sqrt{1-4\left(-7\right)}}{2}$$

Multiply -4 times -7.

$$y=\frac{-\left(-1\right)±\sqrt{1+28}}{2}$$

Add 1 to 28.

$$y=\frac{-\left(-1\right)±\sqrt{29}}{2}$$

The opposite of -1 is 1.

$$y=\frac{1±\sqrt{29}}{2}$$

Now solve the equation y=\frac{1±\sqrt{29}}{2} when ± is plus. Add 1 to \sqrt{29}.

$$y=\frac{\sqrt{29}+1}{2}$$

Now solve the equation y=\frac{1±\sqrt{29}}{2} when ± is minus. Subtract \sqrt{29} from 1.

$$y=\frac{1-\sqrt{29}}{2}$$

The equation is now solved.

$$y=\frac{\sqrt{29}+1}{2} y=\frac{1-\sqrt{29}}{2}$$

Steps for Completing the Square

Use the distributive property to multiply y+2 by y-3 and combine like terms.

$$y^{2}-y-6=1$$

Add 6 to both sides.

$$y^{2}-y=1+6$$

Add 1 and 6 to get 7.

$$y^{2}-y=7$$

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.

$$y^{2}-y+\left(-\frac{1}{2}\right)^{2}=7+\left(-\frac{1}{2}\right)^{2}$$

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

$$y^{2}-y+\frac{1}{4}=7+\frac{1}{4}$$

Add 7 to \frac{1}{4}.

$$y^{2}-y+\frac{1}{4}=\frac{29}{4}$$

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}.

$$\left(y-\frac{1}{2}\right)^{2}=\frac{29}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{1}{2}\right)^{2}}=\sqrt{\frac{29}{4}}$$

Simplify.

$$y-\frac{1}{2}=\frac{\sqrt{29}}{2} y-\frac{1}{2}=-\frac{\sqrt{29}}{2}$$

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

$$y=\frac{\sqrt{29}+1}{2} y=\frac{1-\sqrt{29}}{2}$$