Solve (3-2a)/(2a) - (1-a^(2))/a^(2)

Question

Simplify the expression

\frac{3 a - 2}{2 a ^{2}}

Evaluate

\frac{3 - 2 a}{2 a} - \frac{1 - a ^{2}}{a ^{2}}

Reduce fractions to a common denominator

\frac{( 3 - 2 a ) a}{2 a \times a} - \frac{( 1 - a ^{2} ) \times 2}{a ^{2} \times 2}

Use the commutative property to reorder the terms

\frac{( 3 - 2 a ) a}{2 a \times a} - \frac{( 1 - a ^{2} ) \times 2}{2 a ^{2}}

Multiply the terms

\frac{( 3 - 2 a ) a}{2 a ^{2}} - \frac{( 1 - a ^{2} ) \times 2}{2 a ^{2}}

Write all numerators above the common denominator

\frac{( 3 - 2 a ) a - ( 1 - a ^{2} ) \times 2}{2 a ^{2}}

Multiply the terms

More Steps

Evaluate

(3 - 2 a) a

Apply the distributive property

3 a - 2 a \times a

Multiply the terms

3 a - 2 a^{2}

\frac{3 a - 2 a ^{2} - ( 1 - a ^{2} ) \times 2}{2 a ^{2}}

Multiply the terms

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Evaluate

(1 - a^{2}) \times 2

Apply the distributive property

1 \times 2 - a^{2} \times 2

Any expression multiplied by 1 remains the same

2 - a^{2} \times 2

Use the commutative property to reorder the terms

2 - 2 a^{2}

\frac{3 a - 2 a ^{2} - ( 2 - 2 a ^{2} )}{2 a ^{2}}

Solution

More Steps

Evaluate

3 a - 2 a^{2} - (2 - 2 a^{2})

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

3 a - 2 a^{2} - 2 + 2 a^{2}

The sum of two opposites equals 0

More Steps

Evaluate

- 2 a^{2} + 2 a^{2}

Collect like terms

(- 2 + 2) a^{2}

Add the coefficients

0 \times a^{2}

Calculate

0

3 a + 0 - 2

Remove 0

3 a - 2

\frac{3 a - 2}{2 a ^{2}}

Show Solution

Find the excluded values

a = 0

Evaluate

\frac{3 - 2 a}{2 a} - \frac{1 - a ^{2}}{a ^{2}}

To find the excluded values,set the denominators equal to 0

2 a = 0 a^{2} = 0

Solve the equations

a = 0 a^{2} = 0

The only way a power can be 0 is when the base equals 0

a = 0 a = 0

Solution

a = 0

Show Solution

Find the roots

a = \frac{2}{3}

Alternative Form

a = 0. \dot{6}

Evaluate

\frac{3 - 2 a}{2 a} - \frac{1 - a ^{2}}{a ^{2}}

To find the roots of the expression,set the expression equal to 0

\frac{3 - 2 a}{2 a} - \frac{1 - a ^{2}}{a ^{2}} = 0

Find the domain

More Steps

Evaluate

{2 a \neq = 0 a^{2} \neq = 0

Calculate

{a \neq = 0 a^{2} \neq = 0

The only way a power can not be 0 is when the base not equals 0

{a \neq = 0 a \neq = 0

Find the intersection

a \neq = 0

\frac{3 - 2 a}{2 a} - \frac{1 - a ^{2}}{a ^{2}} = 0, a \neq = 0

Calculate

\frac{3 - 2 a}{2 a} - \frac{1 - a ^{2}}{a ^{2}} = 0

Subtract the terms

More Steps

Simplify

\frac{3 - 2 a}{2 a} - \frac{1 - a ^{2}}{a ^{2}}

Reduce fractions to a common denominator

\frac{( 3 - 2 a ) a}{2 a \times a} - \frac{( 1 - a ^{2} ) \times 2}{a ^{2} \times 2}

Use the commutative property to reorder the terms

\frac{( 3 - 2 a ) a}{2 a \times a} - \frac{( 1 - a ^{2} ) \times 2}{2 a ^{2}}

Multiply the terms

\frac{( 3 - 2 a ) a}{2 a ^{2}} - \frac{( 1 - a ^{2} ) \times 2}{2 a ^{2}}

Write all numerators above the common denominator

\frac{( 3 - 2 a ) a - ( 1 - a ^{2} ) \times 2}{2 a ^{2}}

Multiply the terms

More Steps

Evaluate

(3 - 2 a) a

Apply the distributive property

3 a - 2 a \times a

Multiply the terms

3 a - 2 a^{2}

\frac{3 a - 2 a ^{2} - ( 1 - a ^{2} ) \times 2}{2 a ^{2}}

Multiply the terms

More Steps

Evaluate

(1 - a^{2}) \times 2

Apply the distributive property

1 \times 2 - a^{2} \times 2

Any expression multiplied by 1 remains the same

2 - a^{2} \times 2

Use the commutative property to reorder the terms

2 - 2 a^{2}

\frac{3 a - 2 a ^{2} - ( 2 - 2 a ^{2} )}{2 a ^{2}}

Subtract the terms

More Steps

Evaluate

3 a - 2 a^{2} - (2 - 2 a^{2})

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

3 a - 2 a^{2} - 2 + 2 a^{2}

The sum of two opposites equals 0

3 a + 0 - 2

Remove 0

3 a - 2

\frac{3 a - 2}{2 a ^{2}}

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

Cross multiply

3 a - 2 = 2 a^{2} \times 0

Simplify the equation

3 a - 2 = 0

Move the constant to the right side

3 a = 0 + 2

Removing 0 doesn't change the value,so remove it from the expression

3 a = 2

Divide both sides

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

Divide the numbers

a = \frac{2}{3}

Check if the solution is in the defined range

a = \frac{2}{3}, a \neq = 0

Solution

a = \frac{2}{3}

Alternative Form

a = 0. \dot{6}

Show Solution