Solve (x^2 - 4x^3) (x^2 2x - 5)

Question

Simplify the expression

2 x^{5} - 5 x^{2} - 8 x^{6} + 20 x^{3}

Evaluate

(x^{2} - 4 x^{3}) (x^{2} \times 2 x - 5)

Multiply

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Evaluate

x^{2} \times 2 x

Multiply the terms with the same base by adding their exponents

x^{2 + 1} \times 2

Add the numbers

x^{3} \times 2

Use the commutative property to reorder the terms

2 x^{3}

(x^{2} - 4 x^{3}) (2 x^{3} - 5)

Apply the distributive property

x^{2} \times 2 x^{3} - x^{2} \times 5 - 4 x^{3} \times 2 x^{3} - (- 4 x^{3} \times 5)

Multiply the terms

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Evaluate

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

Use the commutative property to reorder the terms

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

Multiply the terms

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Evaluate

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

Use the product rule a^{n} \times a^{m} = a^{n + m} to simplify the expression

x^{2 + 3}

Add the numbers

x^{5}

2 x^{5}

2 x^{5} - x^{2} \times 5 - 4 x^{3} \times 2 x^{3} - (- 4 x^{3} \times 5)

Use the commutative property to reorder the terms

2 x^{5} - 5 x^{2} - 4 x^{3} \times 2 x^{3} - (- 4 x^{3} \times 5)

Multiply the terms

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Evaluate

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

Multiply the numbers

- 8 x^{3} \times x^{3}

Multiply the terms

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Evaluate

x^{3} \times x^{3}

Use the product rule a^{n} \times a^{m} = a^{n + m} to simplify the expression

x^{3 + 3}

Add the numbers

x^{6}

- 8 x^{6}

2 x^{5} - 5 x^{2} - 8 x^{6} - (- 4 x^{3} \times 5)

Multiply the numbers

2 x^{5} - 5 x^{2} - 8 x^{6} - (- 20 x^{3})

Solution

2 x^{5} - 5 x^{2} - 8 x^{6} + 20 x^{3}

Show Solution

Factor the expression

x^{2} (1 - 4 x) (2 x^{3} - 5)

Evaluate

(x^{2} - 4 x^{3}) (x^{2} \times 2 x - 5)

Multiply

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Evaluate

x^{2} \times 2 x

Multiply the terms with the same base by adding their exponents

x^{2 + 1} \times 2

Add the numbers

x^{3} \times 2

Use the commutative property to reorder the terms

2 x^{3}

(x^{2} - 4 x^{3}) (2 x^{3} - 5)

Solution

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Evaluate

x^{2} - 4 x^{3}

Rewrite the expression

x^{2} - x^{2} \times 4 x

Factor out x^{2} from the expression

x^{2} (1 - 4 x)

x^{2} (1 - 4 x) (2 x^{3} - 5)

Show Solution

Find the roots

x_{1} = 0, x_{2} = \frac{1}{4}, x_{3} = \frac{3 20}{2}

Alternative Form

x_{1} = 0, x_{2} = 0.25, x_{3} \approx 1.357209

Evaluate

(x^{2} - 4 x^{3}) (x^{2} \times 2 x - 5)

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

(x^{2} - 4 x^{3}) (x^{2} \times 2 x - 5) = 0

Multiply

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Multiply the terms

x^{2} \times 2 x

Multiply the terms with the same base by adding their exponents

x^{2 + 1} \times 2

Add the numbers

x^{3} \times 2

Use the commutative property to reorder the terms

2 x^{3}

(x^{2} - 4 x^{3}) (2 x^{3} - 5) = 0

Separate the equation into 2 possible cases

x^{2} - 4 x^{3} = 0 2 x^{3} - 5 = 0

Solve the equation

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Evaluate

x^{2} - 4 x^{3} = 0

Factor the expression

x^{2} (1 - 4 x) = 0

Separate the equation into 2 possible cases

x^{2} = 0 1 - 4 x = 0

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

x = 0 1 - 4 x = 0

Solve the equation

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Evaluate

1 - 4 x = 0

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

- 4 x = 0 - 1

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

- 4 x = - 1

Change the signs on both sides of the equation

4 x = 1

Divide both sides

\frac{4 x}{4} = \frac{1}{4}

Divide the numbers

x = \frac{1}{4}

x = 0 x = \frac{1}{4}

x = 0 x = \frac{1}{4} 2 x^{3} - 5 = 0

Solve the equation

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Evaluate

2 x^{3} - 5 = 0

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

2 x^{3} = 0 + 5

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

2 x^{3} = 5

Divide both sides

\frac{2 x ^{3}}{2} = \frac{5}{2}

Divide the numbers

x^{3} = \frac{5}{2}

Take the 3 -th root on both sides of the equation

3 x^{3} = 3 \frac{5}{2}

Calculate

x = 3 \frac{5}{2}

Simplify the root

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Evaluate

3 \frac{5}{2}

To take a root of a fraction,take the root of the numerator and denominator separately

\frac{3 5}{3 2}

Multiply by the Conjugate

\frac{3 5 \times 3 2 ^{2}}{3 2 \times 3 2 ^{2}}

Simplify

\frac{3 5 \times 3 4}{3 2 \times 3 2 ^{2}}

Multiply the numbers

\frac{3 20}{3 2 \times 3 2 ^{2}}

Multiply the numbers

\frac{3 20}{2}

x = \frac{3 20}{2}

x = 0 x = \frac{1}{4} x = \frac{3 20}{2}

Solution

x_{1} = 0, x_{2} = \frac{1}{4}, x_{3} = \frac{3 20}{2}

Alternative Form

x_{1} = 0, x_{2} = 0.25, x_{3} \approx 1.357209

Show Solution