Solve 7x^2=21x

$$7 x ^ { 2 } = 21 x$$

Solve for x

$$x=3$$

$$x=0$$

Steps Using Factoring

Subtract 21x from both sides.

$$7x^{2}-21x=0$$

Factor out x.

$$x\left(7x-21\right)=0$$

To find equation solutions, solve x=0 and 7x-21=0.

$$x=0 x=3$$

Steps Using the Quadratic Formula

Subtract 21x from both sides.

$$7x^{2}-21x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, -21 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}}}{2\times 7}$$

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

$$x=\frac{-\left(-21\right)±21}{2\times 7}$$

The opposite of -21 is 21.

$$x=\frac{21±21}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{21±21}{14}$$

Now solve the equation x=\frac{21±21}{14} when ± is plus. Add 21 to 21.

$$x=\frac{42}{14}$$

Divide 42 by 14.

$$x=3$$

Now solve the equation x=\frac{21±21}{14} when ± is minus. Subtract 21 from 21.

$$x=\frac{0}{14}$$

Divide 0 by 14.

$$x=0$$

The equation is now solved.

$$x=3 x=0$$

Steps for Completing the Square

Subtract 21x from both sides.

$$7x^{2}-21x=0$$

Divide both sides by 7.

$$\frac{7x^{2}-21x}{7}=\frac{0}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$x^{2}+\left(-\frac{21}{7}\right)x=\frac{0}{7}$$

Divide -21 by 7.

$$x^{2}-3x=\frac{0}{7}$$

Divide 0 by 7.

$$x^{2}-3x=0$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=\frac{9}{4}$$

Factor x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

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

Add \frac{3}{2} to both sides of the equation.

$$x=3 x=0$$