Solve x^2=x+20x

$${ x }^{ 2 } = x+20x$$

Solve for x

$$x=21$$

$$x=0$$

Steps Using Factoring

Combine x and 20x to get 21x.

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

Subtract 21x from both sides.

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

Factor out x.

$$x\left(x-21\right)=0$$

To find equation solutions, solve x=0 and x-21=0.

$$x=0 x=21$$

Steps Using the Quadratic Formula

Combine x and 20x to get 21x.

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

Subtract 21x from both sides.

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

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

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

Take the square root of \left(-21\right)^{2}.

$$x=\frac{-\left(-21\right)±21}{2}$$

The opposite of -21 is 21.

$$x=\frac{21±21}{2}$$

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

$$x=\frac{42}{2}$$

Divide 42 by 2.

$$x=21$$

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

$$x=\frac{0}{2}$$

Divide 0 by 2.

$$x=0$$

The equation is now solved.

$$x=21 x=0$$

Steps for Completing the Square

Combine x and 20x to get 21x.

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

Subtract 21x from both sides.

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

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.

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

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

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

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}.

$$\left(x-\frac{21}{2}\right)^{2}=\frac{441}{4}$$

Take the square root of both sides of the equation.

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

Simplify.

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

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

$$x=21 x=0$$