Solve left(x-5right)left(x-6right)=90

$$\left( x-5 \right) \left( x-6 \right) = 90$$

Solve for x

$$x=-4$$

$$x=15$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-5 by x-6 and combine like terms.

$$x^{2}-11x+30=90$$

Subtract 90 from both sides.

$$x^{2}-11x+30-90=0$$

Subtract 90 from 30 to get -60.

$$x^{2}-11x-60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -11 for b, and -60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-60\right)}}{2}$$

Square -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\left(-60\right)}}{2}$$

Multiply -4 times -60.

$$x=\frac{-\left(-11\right)±\sqrt{121+240}}{2}$$

Add 121 to 240.

$$x=\frac{-\left(-11\right)±\sqrt{361}}{2}$$

Take the square root of 361.

$$x=\frac{-\left(-11\right)±19}{2}$$

The opposite of -11 is 11.

$$x=\frac{11±19}{2}$$

Now solve the equation x=\frac{11±19}{2} when ± is plus. Add 11 to 19.

$$x=\frac{30}{2}$$

Divide 30 by 2.

$$x=15$$

Now solve the equation x=\frac{11±19}{2} when ± is minus. Subtract 19 from 11.

$$x=-\frac{8}{2}$$

Divide -8 by 2.

$$x=-4$$

The equation is now solved.

$$x=15 x=-4$$

Steps for Completing the Square

Use the distributive property to multiply x-5 by x-6 and combine like terms.

$$x^{2}-11x+30=90$$

Subtract 30 from both sides.

$$x^{2}-11x=90-30$$

Subtract 30 from 90 to get 60.

$$x^{2}-11x=60$$

Divide -11, the coefficient of the x term, by 2 to get -\frac{11}{2}. Then add the square of -\frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-11x+\left(-\frac{11}{2}\right)^{2}=60+\left(-\frac{11}{2}\right)^{2}$$

Square -\frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-11x+\frac{121}{4}=60+\frac{121}{4}$$

Add 60 to \frac{121}{4}.

$$x^{2}-11x+\frac{121}{4}=\frac{361}{4}$$

Factor x^{2}-11x+\frac{121}{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}.

$$\left(x-\frac{11}{2}\right)^{2}=\frac{361}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{2}\right)^{2}}=\sqrt{\frac{361}{4}}$$

Simplify.

$$x-\frac{11}{2}=\frac{19}{2} x-\frac{11}{2}=-\frac{19}{2}$$

Add \frac{11}{2} to both sides of the equation.

$$x=15 x=-4$$