Solve 7a^2+14a=7a

$$7 a ^ { 2 } + 14 a = 7 a$$

Solve for a

$$a=-1$$

$$a=0$$

Steps Using the Quadratic Formula

Subtract 7a from both sides.

$$7a^{2}+14a-7a=0$$

Combine 14a and -7a to get 7a.

$$7a^{2}+7a=0$$

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

$$a=\frac{-7±\sqrt{7^{2}}}{2\times 7}$$

Take the square root of 7^{2}.

$$a=\frac{-7±7}{2\times 7}$$

Multiply 2 times 7.

$$a=\frac{-7±7}{14}$$

Now solve the equation a=\frac{-7±7}{14} when ± is plus. Add -7 to 7.

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

Divide 0 by 14.

$$a=0$$

Now solve the equation a=\frac{-7±7}{14} when ± is minus. Subtract 7 from -7.

$$a=-\frac{14}{14}$$

Divide -14 by 14.

$$a=-1$$

The equation is now solved.

$$a=0 a=-1$$

Steps for Completing the Square

Subtract 7a from both sides.

$$7a^{2}+14a-7a=0$$

Combine 14a and -7a to get 7a.

$$7a^{2}+7a=0$$

Divide both sides by 7.

$$\frac{7a^{2}+7a}{7}=\frac{0}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$a^{2}+\frac{7}{7}a=\frac{0}{7}$$

Divide 7 by 7.

$$a^{2}+a=\frac{0}{7}$$

Divide 0 by 7.

$$a^{2}+a=0$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+a+\left(\frac{1}{2}\right)^{2}=\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+a+\frac{1}{4}=\frac{1}{4}$$

Factor a^{2}+a+\frac{1}{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}.

$$\left(a+\frac{1}{2}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{1}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$a+\frac{1}{2}=\frac{1}{2} a+\frac{1}{2}=-\frac{1}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$a=0 a=-1$$