Solve a^2/(a-1)^2=a/7(a-1)

$$\frac { a ^ { 2 } } { ( a - 1 ) ^ { 2 } } = \frac { a } { 7 ( a - 1 ) }$$

Solve for a

$$a=-\frac{1}{6}\approx -0.166666667$$

$$a=0$$

Steps Using Factoring

Variable a cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by 7\left(a-1\right)^{2}, the least common multiple of \left(a-1\right)^{2},7\left(a-1\right).

$$7a^{2}=\left(a-1\right)a$$

Use the distributive property to multiply a-1 by a.

$$7a^{2}=a^{2}-a$$

Subtract a^{2} from both sides.

$$7a^{2}-a^{2}=-a$$

Combine 7a^{2} and -a^{2} to get 6a^{2}.

$$6a^{2}=-a$$

Add a to both sides.

$$6a^{2}+a=0$$

Factor out a.

$$a\left(6a+1\right)=0$$

To find equation solutions, solve a=0 and 6a+1=0.

$$a=0 a=-\frac{1}{6}$$

Steps Using the Quadratic Formula

Variable a cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by 7\left(a-1\right)^{2}, the least common multiple of \left(a-1\right)^{2},7\left(a-1\right).

$$7a^{2}=\left(a-1\right)a$$

Use the distributive property to multiply a-1 by a.

$$7a^{2}=a^{2}-a$$

Subtract a^{2} from both sides.

$$7a^{2}-a^{2}=-a$$

Combine 7a^{2} and -a^{2} to get 6a^{2}.

$$6a^{2}=-a$$

Add a to both sides.

$$6a^{2}+a=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 1 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-1±\sqrt{1^{2}}}{2\times 6}$$

Take the square root of 1^{2}.

$$a=\frac{-1±1}{2\times 6}$$

Multiply 2 times 6.

$$a=\frac{-1±1}{12}$$

Now solve the equation a=\frac{-1±1}{12} when ± is plus. Add -1 to 1.

$$a=\frac{0}{12}$$

Divide 0 by 12.

$$a=0$$

Now solve the equation a=\frac{-1±1}{12} when ± is minus. Subtract 1 from -1.

$$a=-\frac{2}{12}$$

Reduce the fraction \frac{-2}{12} to lowest terms by extracting and canceling out 2.

$$a=-\frac{1}{6}$$

The equation is now solved.

$$a=0 a=-\frac{1}{6}$$

Steps for Completing the Square

Variable a cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by 7\left(a-1\right)^{2}, the least common multiple of \left(a-1\right)^{2},7\left(a-1\right).

$$7a^{2}=\left(a-1\right)a$$

Use the distributive property to multiply a-1 by a.

$$7a^{2}=a^{2}-a$$

Subtract a^{2} from both sides.

$$7a^{2}-a^{2}=-a$$

Combine 7a^{2} and -a^{2} to get 6a^{2}.

$$6a^{2}=-a$$

Add a to both sides.

$$6a^{2}+a=0$$

Divide both sides by 6.

$$\frac{6a^{2}+a}{6}=\frac{0}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$a^{2}+\frac{1}{6}a=\frac{0}{6}$$

Divide 0 by 6.

$$a^{2}+\frac{1}{6}a=0$$

Divide \frac{1}{6}, the coefficient of the x term, by 2 to get \frac{1}{12}. Then add the square of \frac{1}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+\frac{1}{6}a+\left(\frac{1}{12}\right)^{2}=\left(\frac{1}{12}\right)^{2}$$

Square \frac{1}{12} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+\frac{1}{6}a+\frac{1}{144}=\frac{1}{144}$$

Factor a^{2}+\frac{1}{6}a+\frac{1}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+\frac{1}{12}\right)^{2}=\frac{1}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{1}{12}\right)^{2}}=\sqrt{\frac{1}{144}}$$

Simplify.

$$a+\frac{1}{12}=\frac{1}{12} a+\frac{1}{12}=-\frac{1}{12}$$

Subtract \frac{1}{12} from both sides of the equation.

$$a=0 a=-\frac{1}{6}$$