Solve (x-3)(x-2)=20

$$( x - 3 ) ( x - 2 ) = 20$$

Solve for x

$$x=-2$$

$$x=7$$

Steps Using the Quadratic Formula

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

$$x^{2}-5x+6=20$$

Subtract 20 from both sides.

$$x^{2}-5x+6-20=0$$

Subtract 20 from 6 to get -14.

$$x^{2}-5x-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -5 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-14\right)}}{2}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\left(-14\right)}}{2}$$

Multiply -4 times -14.

$$x=\frac{-\left(-5\right)±\sqrt{25+56}}{2}$$

Add 25 to 56.

$$x=\frac{-\left(-5\right)±\sqrt{81}}{2}$$

Take the square root of 81.

$$x=\frac{-\left(-5\right)±9}{2}$$

The opposite of -5 is 5.

$$x=\frac{5±9}{2}$$

Now solve the equation x=\frac{5±9}{2} when ± is plus. Add 5 to 9.

$$x=\frac{14}{2}$$

Divide 14 by 2.

$$x=7$$

Now solve the equation x=\frac{5±9}{2} when ± is minus. Subtract 9 from 5.

$$x=-\frac{4}{2}$$

Divide -4 by 2.

$$x=-2$$

The equation is now solved.

$$x=7 x=-2$$

Steps for Completing the Square

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

$$x^{2}-5x+6=20$$

Subtract 6 from both sides.

$$x^{2}-5x=20-6$$

Subtract 6 from 20 to get 14.

$$x^{2}-5x=14$$

Divide -5, the coefficient of the x term, by 2 to get -\frac{5}{2}. Then add the square of -\frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=14+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-5x+\frac{25}{4}=14+\frac{25}{4}$$

Add 14 to \frac{25}{4}.

$$x^{2}-5x+\frac{25}{4}=\frac{81}{4}$$

Factor x^{2}-5x+\frac{25}{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}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{81}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Simplify.

$$x-\frac{5}{2}=\frac{9}{2} x-\frac{5}{2}=-\frac{9}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$x=7 x=-2$$