Solve (4x^36x^2 - 10x^4) / (2x - 1)

Question

Simplify the expression

\frac{24 x ^{5} - 10 x ^{4}}{2 x - 1}

Evaluate

\frac{4 x ^{3} \times 6 x ^{2} - 10 x ^{4}}{2 x - 1}

Solution

More Steps

Evaluate

4 x^{3} \times 6 x^{2}

Multiply the terms

24 x^{3} \times x^{2}

Multiply the terms with the same base by adding their exponents

24 x^{3 + 2}

Add the numbers

24 x^{5}

\frac{24 x ^{5} - 10 x ^{4}}{2 x - 1}

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Find the excluded values

x = \frac{1}{2}

Evaluate

\frac{4 x ^{3} \times 6 x ^{2} - 10 x ^{4}}{2 x - 1}

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

2 x - 1 = 0

Move the constant to the right-hand side and change its sign

2 x = 0 + 1

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

2 x = 1

Divide both sides

\frac{2 x}{2} = \frac{1}{2}

Solution

x = \frac{1}{2}

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Find the roots

x_{1} = 0, x_{2} = \frac{5}{12}

Alternative Form

x_{1} = 0, x_{2} = 0.41 \dot{6}

Evaluate

\frac{4 x ^{3} \times 6 x ^{2} - 10 x ^{4}}{2 x - 1}

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

\frac{4 x ^{3} \times 6 x ^{2} - 10 x ^{4}}{2 x - 1} = 0

Find the domain

More Steps

Evaluate

2 x - 1 \neq = 0

Move the constant to the right side

2 x \neq = 0 + 1

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

2 x \neq = 1

Divide both sides

\frac{2 x}{2} \neq = \frac{1}{2}

Divide the numbers

x \neq = \frac{1}{2}

\frac{4 x ^{3} \times 6 x ^{2} - 10 x ^{4}}{2 x - 1} = 0, x \neq = \frac{1}{2}

Calculate

\frac{4 x ^{3} \times 6 x ^{2} - 10 x ^{4}}{2 x - 1} = 0

Multiply

More Steps

Multiply the terms

4 x^{3} \times 6 x^{2}

Multiply the terms

24 x^{3} \times x^{2}

Multiply the terms with the same base by adding their exponents

24 x^{3 + 2}

Add the numbers

24 x^{5}

\frac{24 x ^{5} - 10 x ^{4}}{2 x - 1} = 0

Cross multiply

24 x^{5} - 10 x^{4} = (2 x - 1) \times 0

Simplify the equation

24 x^{5} - 10 x^{4} = 0

Factor the expression

2 x^{4} (12 x - 5) = 0

Divide both sides

x^{4} (12 x - 5) = 0

Separate the equation into 2 possible cases

x^{4} = 0 12 x - 5 = 0

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

x = 0 12 x - 5 = 0

Solve the equation

More Steps

Evaluate

12 x - 5 = 0

Move the constant to the right-hand side and change its sign

12 x = 0 + 5

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

12 x = 5

Divide both sides

\frac{12 x}{12} = \frac{5}{12}

Divide the numbers

x = \frac{5}{12}

x = 0 x = \frac{5}{12}

Check if the solution is in the defined range

x = 0 x = \frac{5}{12}, x \neq = \frac{1}{2}

Find the intersection of the solution and the defined range

x = 0 x = \frac{5}{12}

Solution

x_{1} = 0, x_{2} = \frac{5}{12}

Alternative Form

x_{1} = 0, x_{2} = 0.41 \dot{6}

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