Solve (x-8)(x-9)=90

$$(x-8)(x-9)=90$$

Solve for x

$$x=-1$$

$$x=18$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-8 by x-9 and combine like terms.

$$x^{2}-17x+72=90$$

Subtract 90 from both sides.

$$x^{2}-17x+72-90=0$$

Subtract 90 from 72 to get -18.

$$x^{2}-17x-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -17 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\left(-18\right)}}{2}$$

Square -17.

$$x=\frac{-\left(-17\right)±\sqrt{289-4\left(-18\right)}}{2}$$

Multiply -4 times -18.

$$x=\frac{-\left(-17\right)±\sqrt{289+72}}{2}$$

Add 289 to 72.

$$x=\frac{-\left(-17\right)±\sqrt{361}}{2}$$

Take the square root of 361.

$$x=\frac{-\left(-17\right)±19}{2}$$

The opposite of -17 is 17.

$$x=\frac{17±19}{2}$$

Now solve the equation x=\frac{17±19}{2} when ± is plus. Add 17 to 19.

$$x=\frac{36}{2}$$

Divide 36 by 2.

$$x=18$$

Now solve the equation x=\frac{17±19}{2} when ± is minus. Subtract 19 from 17.

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

Divide -2 by 2.

$$x=-1$$

The equation is now solved.

$$x=18 x=-1$$

Steps for Completing the Square

Use the distributive property to multiply x-8 by x-9 and combine like terms.

$$x^{2}-17x+72=90$$

Subtract 72 from both sides.

$$x^{2}-17x=90-72$$

Subtract 72 from 90 to get 18.

$$x^{2}-17x=18$$

Divide -17, the coefficient of the x term, by 2 to get -\frac{17}{2}. Then add the square of -\frac{17}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-17x+\left(-\frac{17}{2}\right)^{2}=18+\left(-\frac{17}{2}\right)^{2}$$

Square -\frac{17}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-17x+\frac{289}{4}=18+\frac{289}{4}$$

Add 18 to \frac{289}{4}.

$$x^{2}-17x+\frac{289}{4}=\frac{361}{4}$$

Factor x^{2}-17x+\frac{289}{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}.

$$\left(x-\frac{17}{2}\right)^{2}=\frac{361}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{17}{2}\right)^{2}}=\sqrt{\frac{361}{4}}$$

Simplify.

$$x-\frac{17}{2}=\frac{19}{2} x-\frac{17}{2}=-\frac{19}{2}$$

Add \frac{17}{2} to both sides of the equation.

$$x=18 x=-1$$