Solve x^2=27x

$${ x }^{ 2 } =27x$$

Solve for x

$$x=27$$

$$x=0$$

Steps Using Factoring

Subtract 27x from both sides.

$$x^{2}-27x=0$$

Factor out x.

$$x\left(x-27\right)=0$$

To find equation solutions, solve x=0 and x-27=0.

$$x=0 x=27$$

Steps Using the Quadratic Formula

Subtract 27x from both sides.

$$x^{2}-27x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -27 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-27\right)±\sqrt{\left(-27\right)^{2}}}{2}$$

Take the square root of \left(-27\right)^{2}.

$$x=\frac{-\left(-27\right)±27}{2}$$

The opposite of -27 is 27.

$$x=\frac{27±27}{2}$$

Now solve the equation x=\frac{27±27}{2} when ± is plus. Add 27 to 27.

$$x=\frac{54}{2}$$

Divide 54 by 2.

$$x=27$$

Now solve the equation x=\frac{27±27}{2} when ± is minus. Subtract 27 from 27.

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

Divide 0 by 2.

$$x=0$$

The equation is now solved.

$$x=27 x=0$$

Steps for Completing the Square

Subtract 27x from both sides.

$$x^{2}-27x=0$$

Divide -27, the coefficient of the x term, by 2 to get -\frac{27}{2}. Then add the square of -\frac{27}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-27x+\left(-\frac{27}{2}\right)^{2}=\left(-\frac{27}{2}\right)^{2}$$

Square -\frac{27}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-27x+\frac{729}{4}=\frac{729}{4}$$

Factor x^{2}-27x+\frac{729}{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}.

$$\left(x-\frac{27}{2}\right)^{2}=\frac{729}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{27}{2}\right)^{2}}=\sqrt{\frac{729}{4}}$$

Simplify.

$$x-\frac{27}{2}=\frac{27}{2} x-\frac{27}{2}=-\frac{27}{2}$$

Add \frac{27}{2} to both sides of the equation.

$$x=27 x=0$$