Solve 4/a-2+(2-a)(2/a-1)/2=3/2

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

Solve for a

$$a = \frac{\sqrt{73} + 19}{18} \approx 1.53022243$$

$$a=\frac{19-\sqrt{73}}{18}\approx 0.580888681$$

Steps Using the Quadratic Formula

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2a, the least common multiple of a,2.

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

Multiply 2 and 4 to get 8.

$$8+2a\left(-2\right)+a\left(2-a\right)\times \frac{2}{a-1}=3a$$

Multiply 2 and -2 to get -4.

$$8-4a+a\left(2-a\right)\times \frac{2}{a-1}=3a$$

Express a\times \frac{2}{a-1} as a single fraction.

$$8-4a+\frac{a\times 2}{a-1}\left(2-a\right)=3a$$

Use the distributive property to multiply \frac{a\times 2}{a-1} by 2-a.

$$8-4a+2\times \frac{a\times 2}{a-1}-\frac{a\times 2}{a-1}a=3a$$

Express 2\times \frac{a\times 2}{a-1} as a single fraction.

$$8-4a+\frac{2a\times 2}{a-1}-\frac{a\times 2}{a-1}a=3a$$

Express \frac{a\times 2}{a-1}a as a single fraction.

$$8-4a+\frac{2a\times 2}{a-1}-\frac{a\times 2a}{a-1}=3a$$

Since \frac{2a\times 2}{a-1} and \frac{a\times 2a}{a-1} have the same denominator, subtract them by subtracting their numerators.

$$8-4a+\frac{2a\times 2-a\times 2a}{a-1}=3a$$

Do the multiplications in 2a\times 2-a\times 2a.

$$8-4a+\frac{4a-2a^{2}}{a-1}=3a$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 8-4a times \frac{a-1}{a-1}.

$$\frac{\left(8-4a\right)\left(a-1\right)}{a-1}+\frac{4a-2a^{2}}{a-1}=3a$$

Since \frac{\left(8-4a\right)\left(a-1\right)}{a-1} and \frac{4a-2a^{2}}{a-1} have the same denominator, add them by adding their numerators.

$$\frac{\left(8-4a\right)\left(a-1\right)+4a-2a^{2}}{a-1}=3a$$

Do the multiplications in \left(8-4a\right)\left(a-1\right)+4a-2a^{2}.

$$\frac{8a-8-4a^{2}+4a+4a-2a^{2}}{a-1}=3a$$

Combine like terms in 8a-8-4a^{2}+4a+4a-2a^{2}.

$$\frac{16a-8-6a^{2}}{a-1}=3a$$

Subtract 3a from both sides.

$$\frac{16a-8-6a^{2}}{a-1}-3a=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -3a times \frac{a-1}{a-1}.

$$\frac{16a-8-6a^{2}}{a-1}+\frac{-3a\left(a-1\right)}{a-1}=0$$

Since \frac{16a-8-6a^{2}}{a-1} and \frac{-3a\left(a-1\right)}{a-1} have the same denominator, add them by adding their numerators.

$$\frac{16a-8-6a^{2}-3a\left(a-1\right)}{a-1}=0$$

Do the multiplications in 16a-8-6a^{2}-3a\left(a-1\right).

$$\frac{16a-8-6a^{2}-3a^{2}+3a}{a-1}=0$$

Combine like terms in 16a-8-6a^{2}-3a^{2}+3a.

$$\frac{19a-8-9a^{2}}{a-1}=0$$

Variable a cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by a-1.

$$19a-8-9a^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-9a^{2}+19a-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 19 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-19±\sqrt{19^{2}-4\left(-9\right)\left(-8\right)}}{2\left(-9\right)}$$

Square 19.

$$a=\frac{-19±\sqrt{361-4\left(-9\right)\left(-8\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$a=\frac{-19±\sqrt{361+36\left(-8\right)}}{2\left(-9\right)}$$

Multiply 36 times -8.

$$a=\frac{-19±\sqrt{361-288}}{2\left(-9\right)}$$

Add 361 to -288.

$$a=\frac{-19±\sqrt{73}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$a=\frac{-19±\sqrt{73}}{-18}$$

Now solve the equation a=\frac{-19±\sqrt{73}}{-18} when ± is plus. Add -19 to \sqrt{73}.

$$a=\frac{\sqrt{73}-19}{-18}$$

Divide -19+\sqrt{73} by -18.

$$a=\frac{19-\sqrt{73}}{18}$$

Now solve the equation a=\frac{-19±\sqrt{73}}{-18} when ± is minus. Subtract \sqrt{73} from -19.

$$a=\frac{-\sqrt{73}-19}{-18}$$

Divide -19-\sqrt{73} by -18.

$$a=\frac{\sqrt{73}+19}{18}$$

The equation is now solved.

$$a=\frac{19-\sqrt{73}}{18} a=\frac{\sqrt{73}+19}{18}$$

Steps for Completing the Square

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2a, the least common multiple of a,2.

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

Multiply 2 and 4 to get 8.

$$8+2a\left(-2\right)+a\left(2-a\right)\times \frac{2}{a-1}=3a$$

Multiply 2 and -2 to get -4.

$$8-4a+a\left(2-a\right)\times \frac{2}{a-1}=3a$$

Express a\times \frac{2}{a-1} as a single fraction.

$$8-4a+\frac{a\times 2}{a-1}\left(2-a\right)=3a$$

Use the distributive property to multiply \frac{a\times 2}{a-1} by 2-a.

$$8-4a+2\times \frac{a\times 2}{a-1}-\frac{a\times 2}{a-1}a=3a$$

Express 2\times \frac{a\times 2}{a-1} as a single fraction.

$$8-4a+\frac{2a\times 2}{a-1}-\frac{a\times 2}{a-1}a=3a$$

Express \frac{a\times 2}{a-1}a as a single fraction.

$$8-4a+\frac{2a\times 2}{a-1}-\frac{a\times 2a}{a-1}=3a$$

Since \frac{2a\times 2}{a-1} and \frac{a\times 2a}{a-1} have the same denominator, subtract them by subtracting their numerators.

$$8-4a+\frac{2a\times 2-a\times 2a}{a-1}=3a$$

Do the multiplications in 2a\times 2-a\times 2a.

$$8-4a+\frac{4a-2a^{2}}{a-1}=3a$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 8-4a times \frac{a-1}{a-1}.

$$\frac{\left(8-4a\right)\left(a-1\right)}{a-1}+\frac{4a-2a^{2}}{a-1}=3a$$

Since \frac{\left(8-4a\right)\left(a-1\right)}{a-1} and \frac{4a-2a^{2}}{a-1} have the same denominator, add them by adding their numerators.

$$\frac{\left(8-4a\right)\left(a-1\right)+4a-2a^{2}}{a-1}=3a$$

Do the multiplications in \left(8-4a\right)\left(a-1\right)+4a-2a^{2}.

$$\frac{8a-8-4a^{2}+4a+4a-2a^{2}}{a-1}=3a$$

Combine like terms in 8a-8-4a^{2}+4a+4a-2a^{2}.

$$\frac{16a-8-6a^{2}}{a-1}=3a$$

Subtract 3a from both sides.

$$\frac{16a-8-6a^{2}}{a-1}-3a=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -3a times \frac{a-1}{a-1}.

$$\frac{16a-8-6a^{2}}{a-1}+\frac{-3a\left(a-1\right)}{a-1}=0$$

Since \frac{16a-8-6a^{2}}{a-1} and \frac{-3a\left(a-1\right)}{a-1} have the same denominator, add them by adding their numerators.

$$\frac{16a-8-6a^{2}-3a\left(a-1\right)}{a-1}=0$$

Do the multiplications in 16a-8-6a^{2}-3a\left(a-1\right).

$$\frac{16a-8-6a^{2}-3a^{2}+3a}{a-1}=0$$

Combine like terms in 16a-8-6a^{2}-3a^{2}+3a.

$$\frac{19a-8-9a^{2}}{a-1}=0$$

Variable a cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by a-1.

$$19a-8-9a^{2}=0$$

Add 8 to both sides. Anything plus zero gives itself.

$$19a-9a^{2}=8$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-9a^{2}+19a=8$$

Divide both sides by -9.

$$\frac{-9a^{2}+19a}{-9}=\frac{8}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$a^{2}+\frac{19}{-9}a=\frac{8}{-9}$$

Divide 19 by -9.

$$a^{2}-\frac{19}{9}a=\frac{8}{-9}$$

Divide 8 by -9.

$$a^{2}-\frac{19}{9}a=-\frac{8}{9}$$

Divide -\frac{19}{9}, the coefficient of the x term, by 2 to get -\frac{19}{18}. Then add the square of -\frac{19}{18} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{19}{9}a+\left(-\frac{19}{18}\right)^{2}=-\frac{8}{9}+\left(-\frac{19}{18}\right)^{2}$$

Square -\frac{19}{18} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{19}{9}a+\frac{361}{324}=-\frac{8}{9}+\frac{361}{324}$$

Add -\frac{8}{9} to \frac{361}{324} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}-\frac{19}{9}a+\frac{361}{324}=\frac{73}{324}$$

Factor a^{2}-\frac{19}{9}a+\frac{361}{324}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{19}{18}\right)^{2}=\frac{73}{324}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{19}{18}\right)^{2}}=\sqrt{\frac{73}{324}}$$

Simplify.

$$a-\frac{19}{18}=\frac{\sqrt{73}}{18} a-\frac{19}{18}=-\frac{\sqrt{73}}{18}$$

Add \frac{19}{18} to both sides of the equation.

$$a=\frac{\sqrt{73}+19}{18} a=\frac{19-\sqrt{73}}{18}$$