Solve x(x-10)-11(x-10)=0

$$x ( x - 10 ) - 11 ( x - 10 ) = 0$$

Solve for x

$$x=11$$

$$x=10$$

Steps Using Factoring

Use the distributive property to multiply x by x-10.

$$x^{2}-10x-11\left(x-10\right)=0$$

Use the distributive property to multiply -11 by x-10.

$$x^{2}-10x-11x+110=0$$

Combine -10x and -11x to get -21x.

$$x^{2}-21x+110=0$$

To solve the equation, factor x^{2}-21x+110 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-21 ab=110$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 110.

$$-1,-110 -2,-55 -5,-22 -10,-11$$

Calculate the sum for each pair.

$$-1-110=-111 -2-55=-57 -5-22=-27 -10-11=-21$$

The solution is the pair that gives sum -21.

$$a=-11 b=-10$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-11\right)\left(x-10\right)$$

To find equation solutions, solve x-11=0 and x-10=0.

$$x=11 x=10$$

Steps Using Factoring By Grouping

Use the distributive property to multiply x by x-10.

$$x^{2}-10x-11\left(x-10\right)=0$$

Use the distributive property to multiply -11 by x-10.

$$x^{2}-10x-11x+110=0$$

Combine -10x and -11x to get -21x.

$$x^{2}-21x+110=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+110. To find a and b, set up a system to be solved.

$$a+b=-21 ab=1\times 110=110$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 110.

$$-1,-110 -2,-55 -5,-22 -10,-11$$

Calculate the sum for each pair.

$$-1-110=-111 -2-55=-57 -5-22=-27 -10-11=-21$$

The solution is the pair that gives sum -21.

$$a=-11 b=-10$$

Rewrite x^{2}-21x+110 as \left(x^{2}-11x\right)+\left(-10x+110\right).

$$\left(x^{2}-11x\right)+\left(-10x+110\right)$$

Factor out x in the first and -10 in the second group.

$$x\left(x-11\right)-10\left(x-11\right)$$

Factor out common term x-11 by using distributive property.

$$\left(x-11\right)\left(x-10\right)$$

To find equation solutions, solve x-11=0 and x-10=0.

$$x=11 x=10$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by x-10.

$$x^{2}-10x-11\left(x-10\right)=0$$

Use the distributive property to multiply -11 by x-10.

$$x^{2}-10x-11x+110=0$$

Combine -10x and -11x to get -21x.

$$x^{2}-21x+110=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -21 for b, and 110 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\times 110}}{2}$$

Square -21.

$$x=\frac{-\left(-21\right)±\sqrt{441-4\times 110}}{2}$$

Multiply -4 times 110.

$$x=\frac{-\left(-21\right)±\sqrt{441-440}}{2}$$

Add 441 to -440.

$$x=\frac{-\left(-21\right)±\sqrt{1}}{2}$$

Take the square root of 1.

$$x=\frac{-\left(-21\right)±1}{2}$$

The opposite of -21 is 21.

$$x=\frac{21±1}{2}$$

Now solve the equation x=\frac{21±1}{2} when ± is plus. Add 21 to 1.

$$x=\frac{22}{2}$$

Divide 22 by 2.

$$x=11$$

Now solve the equation x=\frac{21±1}{2} when ± is minus. Subtract 1 from 21.

$$x=\frac{20}{2}$$

Divide 20 by 2.

$$x=10$$

The equation is now solved.

$$x=11 x=10$$

Steps for Completing the Square

Use the distributive property to multiply x by x-10.

$$x^{2}-10x-11\left(x-10\right)=0$$

Use the distributive property to multiply -11 by x-10.

$$x^{2}-10x-11x+110=0$$

Combine -10x and -11x to get -21x.

$$x^{2}-21x+110=0$$

Subtract 110 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-21x=-110$$

Divide -21, the coefficient of the x term, by 2 to get -\frac{21}{2}. Then add the square of -\frac{21}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-21x+\left(-\frac{21}{2}\right)^{2}=-110+\left(-\frac{21}{2}\right)^{2}$$

Square -\frac{21}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-21x+\frac{441}{4}=-110+\frac{441}{4}$$

Add -110 to \frac{441}{4}.

$$x^{2}-21x+\frac{441}{4}=\frac{1}{4}$$

Factor x^{2}-21x+\frac{441}{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}.

$$\left(x-\frac{21}{2}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{21}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$x-\frac{21}{2}=\frac{1}{2} x-\frac{21}{2}=-\frac{1}{2}$$

Add \frac{21}{2} to both sides of the equation.

$$x=11 x=10$$