Solve (x-1)(x-2)=30

$$(x-1)(x-2)=30$$

Solve for x

$$x=-4$$

$$x=7$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-1 by x-2 and combine like terms.

$$x^{2}-3x+2=30$$

Subtract 30 from both sides.

$$x^{2}-3x+2-30=0$$

Subtract 30 from 2 to get -28.

$$x^{2}-3x-28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and -28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-28\right)}}{2}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-28\right)}}{2}$$

Multiply -4 times -28.

$$x=\frac{-\left(-3\right)±\sqrt{9+112}}{2}$$

Add 9 to 112.

$$x=\frac{-\left(-3\right)±\sqrt{121}}{2}$$

Take the square root of 121.

$$x=\frac{-\left(-3\right)±11}{2}$$

The opposite of -3 is 3.

$$x=\frac{3±11}{2}$$

Now solve the equation x=\frac{3±11}{2} when ± is plus. Add 3 to 11.

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

Divide 14 by 2.

$$x=7$$

Now solve the equation x=\frac{3±11}{2} when ± is minus. Subtract 11 from 3.

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

Divide -8 by 2.

$$x=-4$$

The equation is now solved.

$$x=7 x=-4$$

Steps for Completing the Square

Use the distributive property to multiply x-1 by x-2 and combine like terms.

$$x^{2}-3x+2=30$$

Subtract 2 from both sides.

$$x^{2}-3x=30-2$$

Subtract 2 from 30 to get 28.

$$x^{2}-3x=28$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=28+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=28+\frac{9}{4}$$

Add 28 to \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{121}{4}$$

Factor x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{11}{2} x-\frac{3}{2}=-\frac{11}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$x=7 x=-4$$