Solve (x-11)(2x+1)=(x-11)

$$(x-11)(2x+1)=(x-11)$$

Solve for x

$$x=11$$

$$x=0$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-11 by 2x+1 and combine like terms.

$$2x^{2}-21x-11=x-11$$

Subtract x from both sides.

$$2x^{2}-21x-11-x=-11$$

Combine -21x and -x to get -22x.

$$2x^{2}-22x-11=-11$$

Add 11 to both sides.

$$2x^{2}-22x-11+11=0$$

Add -11 and 11 to get 0.

$$2x^{2}-22x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -22 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}}}{2\times 2}$$

Take the square root of \left(-22\right)^{2}.

$$x=\frac{-\left(-22\right)±22}{2\times 2}$$

The opposite of -22 is 22.

$$x=\frac{22±22}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{22±22}{4}$$

Now solve the equation x=\frac{22±22}{4} when ± is plus. Add 22 to 22.

$$x=\frac{44}{4}$$

Divide 44 by 4.

$$x=11$$

Now solve the equation x=\frac{22±22}{4} when ± is minus. Subtract 22 from 22.

$$x=\frac{0}{4}$$

Divide 0 by 4.

$$x=0$$

The equation is now solved.

$$x=11 x=0$$

Steps for Completing the Square

Use the distributive property to multiply x-11 by 2x+1 and combine like terms.

$$2x^{2}-21x-11=x-11$$

Subtract x from both sides.

$$2x^{2}-21x-11-x=-11$$

Combine -21x and -x to get -22x.

$$2x^{2}-22x-11=-11$$

Add 11 to both sides.

$$2x^{2}-22x=-11+11$$

Add -11 and 11 to get 0.

$$2x^{2}-22x=0$$

Divide both sides by 2.

$$\frac{2x^{2}-22x}{2}=\frac{0}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{22}{2}\right)x=\frac{0}{2}$$

Divide -22 by 2.

$$x^{2}-11x=\frac{0}{2}$$

Divide 0 by 2.

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

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.

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

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

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

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

$$x=11 x=0$$