Solve (x-1)(x-6)-144=0

$$( x - 1 ) ( x - 6 ) - 144 = 0$$

Solve for x

$$x = \frac{\sqrt{601} + 7}{2} \approx 15.757650672$$

$$x=\frac{7-\sqrt{601}}{2}\approx -8.757650672$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-1 by x-6 and combine like terms.

$$x^{2}-7x+6-144=0$$

Subtract 144 from 6 to get -138.

$$x^{2}-7x-138=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -7 for b, and -138 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-138\right)}}{2}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-138\right)}}{2}$$

Multiply -4 times -138.

$$x=\frac{-\left(-7\right)±\sqrt{49+552}}{2}$$

Add 49 to 552.

$$x=\frac{-\left(-7\right)±\sqrt{601}}{2}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{601}}{2}$$

Now solve the equation x=\frac{7±\sqrt{601}}{2} when ± is plus. Add 7 to \sqrt{601}.

$$x=\frac{\sqrt{601}+7}{2}$$

Now solve the equation x=\frac{7±\sqrt{601}}{2} when ± is minus. Subtract \sqrt{601} from 7.

$$x=\frac{7-\sqrt{601}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{601}+7}{2} x=\frac{7-\sqrt{601}}{2}$$

Steps for Completing the Square

Use the distributive property to multiply x-1 by x-6 and combine like terms.

$$x^{2}-7x+6-144=0$$

Subtract 144 from 6 to get -138.

$$x^{2}-7x-138=0$$

Add 138 to both sides. Anything plus zero gives itself.

$$x^{2}-7x=138$$

Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=138+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-7x+\frac{49}{4}=138+\frac{49}{4}$$

Add 138 to \frac{49}{4}.

$$x^{2}-7x+\frac{49}{4}=\frac{601}{4}$$

Factor x^{2}-7x+\frac{49}{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}.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{601}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{601}{4}}$$

Simplify.

$$x-\frac{7}{2}=\frac{\sqrt{601}}{2} x-\frac{7}{2}=-\frac{\sqrt{601}}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$x=\frac{\sqrt{601}+7}{2} x=\frac{7-\sqrt{601}}{2}$$