Solve a/2=1/2a/2-frac{3{2}a}

$$\frac { a } { 2 } = \frac { \frac { 1 } { 2 } a } { 2 - \frac { 3 } { 2 } a }$$

Solve for a

$$a=\frac{2}{3}\approx 0.666666667$$

$$a=0$$

Steps Using Factoring

Factor the expressions that are not already factored in \frac{\frac{1}{2}a}{2-\frac{3}{2}a}.

$$\frac{a}{2}=\frac{\frac{1}{2}a}{\frac{1}{2}\left(-3a+4\right)}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{a}{2}=\frac{a}{\left(\frac{1}{2}\right)^{0}\left(-3a+4\right)}$$

Calculate \frac{1}{2} to the power of 0 and get 1.

$$\frac{a}{2}=\frac{a}{1\left(-3a+4\right)}$$

Use the distributive property to multiply 1 by -3a+4.

$$\frac{a}{2}=\frac{a}{-3a+4}$$

Subtract \frac{a}{-3a+4} from both sides.

$$\frac{a}{2}-\frac{a}{-3a+4}=0$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and -3a+4 is 2\left(-3a+4\right). Multiply \frac{a}{2} times \frac{-3a+4}{-3a+4}. Multiply \frac{a}{-3a+4} times \frac{2}{2}.

$$\frac{a\left(-3a+4\right)}{2\left(-3a+4\right)}-\frac{2a}{2\left(-3a+4\right)}=0$$

Since \frac{a\left(-3a+4\right)}{2\left(-3a+4\right)} and \frac{2a}{2\left(-3a+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{a\left(-3a+4\right)-2a}{2\left(-3a+4\right)}=0$$

Do the multiplications in a\left(-3a+4\right)-2a.

$$\frac{-3a^{2}+4a-2a}{2\left(-3a+4\right)}=0$$

Combine like terms in -3a^{2}+4a-2a.

$$\frac{-3a^{2}+2a}{2\left(-3a+4\right)}=0$$

Variable a cannot be equal to \frac{4}{3} since division by zero is not defined. Multiply both sides of the equation by 2\left(-3a+4\right).

$$-3a^{2}+2a=0$$

Factor out a.

$$a\left(-3a+2\right)=0$$

To find equation solutions, solve a=0 and -3a+2=0.

$$a=0 a=\frac{2}{3}$$

Steps Using the Quadratic Formula

Factor the expressions that are not already factored in \frac{\frac{1}{2}a}{2-\frac{3}{2}a}.

$$\frac{a}{2}=\frac{\frac{1}{2}a}{\frac{1}{2}\left(-3a+4\right)}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{a}{2}=\frac{a}{\left(\frac{1}{2}\right)^{0}\left(-3a+4\right)}$$

Calculate \frac{1}{2} to the power of 0 and get 1.

$$\frac{a}{2}=\frac{a}{1\left(-3a+4\right)}$$

Use the distributive property to multiply 1 by -3a+4.

$$\frac{a}{2}=\frac{a}{-3a+4}$$

Subtract \frac{a}{-3a+4} from both sides.

$$\frac{a}{2}-\frac{a}{-3a+4}=0$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and -3a+4 is 2\left(-3a+4\right). Multiply \frac{a}{2} times \frac{-3a+4}{-3a+4}. Multiply \frac{a}{-3a+4} times \frac{2}{2}.

$$\frac{a\left(-3a+4\right)}{2\left(-3a+4\right)}-\frac{2a}{2\left(-3a+4\right)}=0$$

Since \frac{a\left(-3a+4\right)}{2\left(-3a+4\right)} and \frac{2a}{2\left(-3a+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{a\left(-3a+4\right)-2a}{2\left(-3a+4\right)}=0$$

Do the multiplications in a\left(-3a+4\right)-2a.

$$\frac{-3a^{2}+4a-2a}{2\left(-3a+4\right)}=0$$

Combine like terms in -3a^{2}+4a-2a.

$$\frac{-3a^{2}+2a}{2\left(-3a+4\right)}=0$$

Variable a cannot be equal to \frac{4}{3} since division by zero is not defined. Multiply both sides of the equation by 2\left(-3a+4\right).

$$-3a^{2}+2a=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 2 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-2±\sqrt{2^{2}}}{2\left(-3\right)}$$

Take the square root of 2^{2}.

$$a=\frac{-2±2}{2\left(-3\right)}$$

Multiply 2 times -3.

$$a=\frac{-2±2}{-6}$$

Now solve the equation a=\frac{-2±2}{-6} when ± is plus. Add -2 to 2.

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

Divide 0 by -6.

$$a=0$$

Now solve the equation a=\frac{-2±2}{-6} when ± is minus. Subtract 2 from -2.

$$a=-\frac{4}{-6}$$

Reduce the fraction \frac{-4}{-6} to lowest terms by extracting and canceling out 2.

$$a=\frac{2}{3}$$

The equation is now solved.

$$a=0 a=\frac{2}{3}$$

Steps for Completing the Square

Factor the expressions that are not already factored in \frac{\frac{1}{2}a}{2-\frac{3}{2}a}.

$$\frac{a}{2}=\frac{\frac{1}{2}a}{\frac{1}{2}\left(-3a+4\right)}$$

To divide powers of the same base, subtract the numerator's exponent from the denominator's exponent.

$$\frac{a}{2}=\frac{a}{\left(\frac{1}{2}\right)^{0}\left(-3a+4\right)}$$

Calculate \frac{1}{2} to the power of 0 and get 1.

$$\frac{a}{2}=\frac{a}{1\left(-3a+4\right)}$$

Use the distributive property to multiply 1 by -3a+4.

$$\frac{a}{2}=\frac{a}{-3a+4}$$

Subtract \frac{a}{-3a+4} from both sides.

$$\frac{a}{2}-\frac{a}{-3a+4}=0$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and -3a+4 is 2\left(-3a+4\right). Multiply \frac{a}{2} times \frac{-3a+4}{-3a+4}. Multiply \frac{a}{-3a+4} times \frac{2}{2}.

$$\frac{a\left(-3a+4\right)}{2\left(-3a+4\right)}-\frac{2a}{2\left(-3a+4\right)}=0$$

Since \frac{a\left(-3a+4\right)}{2\left(-3a+4\right)} and \frac{2a}{2\left(-3a+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{a\left(-3a+4\right)-2a}{2\left(-3a+4\right)}=0$$

Do the multiplications in a\left(-3a+4\right)-2a.

$$\frac{-3a^{2}+4a-2a}{2\left(-3a+4\right)}=0$$

Combine like terms in -3a^{2}+4a-2a.

$$\frac{-3a^{2}+2a}{2\left(-3a+4\right)}=0$$

Variable a cannot be equal to \frac{4}{3} since division by zero is not defined. Multiply both sides of the equation by 2\left(-3a+4\right).

$$-3a^{2}+2a=0$$

Divide both sides by -3.

$$\frac{-3a^{2}+2a}{-3}=\frac{0}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$a^{2}+\frac{2}{-3}a=\frac{0}{-3}$$

Divide 2 by -3.

$$a^{2}-\frac{2}{3}a=\frac{0}{-3}$$

Divide 0 by -3.

$$a^{2}-\frac{2}{3}a=0$$

Divide -\frac{2}{3}, the coefficient of the x term, by 2 to get -\frac{1}{3}. Then add the square of -\frac{1}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{2}{3}a+\left(-\frac{1}{3}\right)^{2}=\left(-\frac{1}{3}\right)^{2}$$

Square -\frac{1}{3} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{2}{3}a+\frac{1}{9}=\frac{1}{9}$$

Factor a^{2}-\frac{2}{3}a+\frac{1}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{1}{3}\right)^{2}=\frac{1}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{1}{3}\right)^{2}}=\sqrt{\frac{1}{9}}$$

Simplify.

$$a-\frac{1}{3}=\frac{1}{3} a-\frac{1}{3}=-\frac{1}{3}$$

Add \frac{1}{3} to both sides of the equation.

$$a=\frac{2}{3} a=0$$