Solve (x-1)(x-2)=(x-1)(3x-10)

$$( x - 1 ) ( x - 2 ) = ( x - 1 ) ( 3 x - 10 )$$

Solve for x

$$x=4$$

$$x=1$$

Steps Using the Quadratic Formula

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

$$x^{2}-3x+2=\left(x-1\right)\left(3x-10\right)$$

Use the distributive property to multiply x-1 by 3x-10 and combine like terms.

$$x^{2}-3x+2=3x^{2}-13x+10$$

Subtract 3x^{2} from both sides.

$$x^{2}-3x+2-3x^{2}=-13x+10$$

Combine x^{2} and -3x^{2} to get -2x^{2}.

$$-2x^{2}-3x+2=-13x+10$$

Add 13x to both sides.

$$-2x^{2}-3x+2+13x=10$$

Combine -3x and 13x to get 10x.

$$-2x^{2}+10x+2=10$$

Subtract 10 from both sides.

$$-2x^{2}+10x+2-10=0$$

Subtract 10 from 2 to get -8.

$$-2x^{2}+10x-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 10 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-2\right)\left(-8\right)}}{2\left(-2\right)}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\left(-2\right)\left(-8\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-10±\sqrt{100+8\left(-8\right)}}{2\left(-2\right)}$$

Multiply 8 times -8.

$$x=\frac{-10±\sqrt{100-64}}{2\left(-2\right)}$$

Add 100 to -64.

$$x=\frac{-10±\sqrt{36}}{2\left(-2\right)}$$

Take the square root of 36.

$$x=\frac{-10±6}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-10±6}{-4}$$

Now solve the equation x=\frac{-10±6}{-4} when ± is plus. Add -10 to 6.

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

Divide -4 by -4.

$$x=1$$

Now solve the equation x=\frac{-10±6}{-4} when ± is minus. Subtract 6 from -10.

$$x=-\frac{16}{-4}$$

Divide -16 by -4.

$$x=4$$

The equation is now solved.

$$x=1 x=4$$

Steps for Completing the Square

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

$$x^{2}-3x+2=\left(x-1\right)\left(3x-10\right)$$

Use the distributive property to multiply x-1 by 3x-10 and combine like terms.

$$x^{2}-3x+2=3x^{2}-13x+10$$

Subtract 3x^{2} from both sides.

$$x^{2}-3x+2-3x^{2}=-13x+10$$

Combine x^{2} and -3x^{2} to get -2x^{2}.

$$-2x^{2}-3x+2=-13x+10$$

Add 13x to both sides.

$$-2x^{2}-3x+2+13x=10$$

Combine -3x and 13x to get 10x.

$$-2x^{2}+10x+2=10$$

Subtract 2 from both sides.

$$-2x^{2}+10x=10-2$$

Subtract 2 from 10 to get 8.

$$-2x^{2}+10x=8$$

Divide both sides by -2.

$$\frac{-2x^{2}+10x}{-2}=\frac{8}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{10}{-2}x=\frac{8}{-2}$$

Divide 10 by -2.

$$x^{2}-5x=\frac{8}{-2}$$

Divide 8 by -2.

$$x^{2}-5x=-4$$

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.

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

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

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

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

$$x^{2}-5x+\frac{25}{4}=\frac{9}{4}$$

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}.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$x-\frac{5}{2}=\frac{3}{2} x-\frac{5}{2}=-\frac{3}{2}$$

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

$$x=4 x=1$$