Solve (4x^(2)-9x x^(3) 12) (-2x^(3) 6x -16)

Question

Simplify the expression

- 48 x^{6} - 64 x^{2} + 1296 x^{8} + 1728 x^{4}

Evaluate

(4 x^{2} - 9 x \times x^{3} \times 12) (- 2 x^{3} \times 6 x - 16)

Multiply

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Multiply the terms

9 x \times x^{3} \times 12

Multiply the terms

108 x \times x^{3}

Multiply the terms with the same base by adding their exponents

108 x^{1 + 3}

Add the numbers

108 x^{4}

(4 x^{2} - 108 x^{4}) (- 2 x^{3} \times 6 x - 16)

Multiply

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Multiply the terms

- 2 x^{3} \times 6 x

Multiply the terms

- 12 x^{3} \times x

Multiply the terms with the same base by adding their exponents

- 12 x^{3 + 1}

Add the numbers

- 12 x^{4}

(4 x^{2} - 108 x^{4}) (- 12 x^{4} - 16)

Apply the distributive property

4 x^{2} (- 12 x^{4}) - 4 x^{2} \times 16 - 108 x^{4} (- 12 x^{4}) - (- 108 x^{4} \times 16)

Multiply the terms

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Evaluate

4 x^{2} (- 12 x^{4})

Multiply the numbers

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Evaluate

4 (- 12)

Multiplying or dividing an odd number of negative terms equals a negative

- 4 \times 12

Multiply the numbers

- 48

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

Multiply the terms

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Evaluate

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

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

x^{2 + 4}

Add the numbers

x^{6}

- 48 x^{6}

- 48 x^{6} - 4 x^{2} \times 16 - 108 x^{4} (- 12 x^{4}) - (- 108 x^{4} \times 16)

Multiply the numbers

- 48 x^{6} - 64 x^{2} - 108 x^{4} (- 12 x^{4}) - (- 108 x^{4} \times 16)

Multiply the terms

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Evaluate

- 108 x^{4} (- 12 x^{4})

Multiply the numbers

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Evaluate

- 108 (- 12)

Multiplying or dividing an even number of negative terms equals a positive

108 \times 12

Multiply the numbers

1296

1296 x^{4} \times x^{4}

Multiply the terms

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Evaluate

x^{4} \times x^{4}

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

x^{4 + 4}

Add the numbers

x^{8}

1296 x^{8}

- 48 x^{6} - 64 x^{2} + 1296 x^{8} - (- 108 x^{4} \times 16)

Multiply the numbers

- 48 x^{6} - 64 x^{2} + 1296 x^{8} - (- 1728 x^{4})

Solution

- 48 x^{6} - 64 x^{2} + 1296 x^{8} + 1728 x^{4}

Show Solution

Factor the expression

- 16 x^{2} (1 - 27 x^{2}) (3 x^{4} + 4)

Evaluate

(4 x^{2} - 9 x \times x^{3} \times 12) (- 2 x^{3} \times 6 x - 16)

Multiply

More Steps

Multiply the terms

9 x \times x^{3} \times 12

Multiply the terms

108 x \times x^{3}

Multiply the terms with the same base by adding their exponents

108 x^{1 + 3}

Add the numbers

108 x^{4}

(4 x^{2} - 108 x^{4}) (- 2 x^{3} \times 6 x - 16)

Multiply

More Steps

Multiply the terms

- 2 x^{3} \times 6 x

Multiply the terms

- 12 x^{3} \times x

Multiply the terms with the same base by adding their exponents

- 12 x^{3 + 1}

Add the numbers

- 12 x^{4}

(4 x^{2} - 108 x^{4}) (- 12 x^{4} - 16)

Factor the expression

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Evaluate

4 x^{2} - 108 x^{4}

Rewrite the expression

4 x^{2} - 4 x^{2} \times 27 x^{2}

Factor out 4 x^{2} from the expression

4 x^{2} (1 - 27 x^{2})

4 x^{2} (1 - 27 x^{2}) (- 12 x^{4} - 16)

Factor the expression

4 x^{2} (1 - 27 x^{2}) (- 4) (3 x^{4} + 4)

Solution

- 16 x^{2} (1 - 27 x^{2}) (3 x^{4} + 4)

Show Solution

Find the roots

x_{1} = - \frac{4 27}{3} - \frac{4 27}{3} i, x_{2} = \frac{4 27}{3} + \frac{4 27}{3} i, x_{3} = - \frac{3}{9}, x_{4} = 0, x_{5} = \frac{3}{9}

Alternative Form

x_{1} \approx - 0.759836 - 0.759836 i, x_{2} \approx 0.759836 + 0.759836 i, x_{3} \approx - 0.19245, x_{4} = 0, x_{5} \approx 0.19245

Evaluate

(4 x^{2} - 9 x \times x^{3} \times 12) (- 2 x^{3} \times 6 x - 16)

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

(4 x^{2} - 9 x \times x^{3} \times 12) (- 2 x^{3} \times 6 x - 16) = 0

Multiply

More Steps

Multiply the terms

9 x \times x^{3} \times 12

Multiply the terms

108 x \times x^{3}

Multiply the terms with the same base by adding their exponents

108 x^{1 + 3}

Add the numbers

108 x^{4}

(4 x^{2} - 108 x^{4}) (- 2 x^{3} \times 6 x - 16) = 0

Multiply

More Steps

Multiply the terms

- 2 x^{3} \times 6 x

Multiply the terms

- 12 x^{3} \times x

Multiply the terms with the same base by adding their exponents

- 12 x^{3 + 1}

Add the numbers

- 12 x^{4}

(4 x^{2} - 108 x^{4}) (- 12 x^{4} - 16) = 0

Separate the equation into 2 possible cases

4 x^{2} - 108 x^{4} = 0 - 12 x^{4} - 16 = 0

Solve the equation

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Evaluate

4 x^{2} - 108 x^{4} = 0

Factor the expression

4 x^{2} (1 - 27 x^{2}) = 0

Divide both sides

x^{2} (1 - 27 x^{2}) = 0

Separate the equation into 2 possible cases

x^{2} = 0 1 - 27 x^{2} = 0

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

x = 0 1 - 27 x^{2} = 0

Solve the equation

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Evaluate

1 - 27 x^{2} = 0

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

- 27 x^{2} = 0 - 1

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

- 27 x^{2} = - 1

Change the signs on both sides of the equation

27 x^{2} = 1

Divide both sides

\frac{27 x ^{2}}{27} = \frac{1}{27}

Divide the numbers

x^{2} = \frac{1}{27}

Take the root of both sides of the equation and remember to use both positive and negative roots

x = \pm \frac{1}{27}

Simplify the expression

x = \pm \frac{3}{9}

Separate the equation into 2 possible cases

x = \frac{3}{9} x = - \frac{3}{9}

x = 0 x = \frac{3}{9} x = - \frac{3}{9}

x = 0 x = \frac{3}{9} x = - \frac{3}{9} - 12 x^{4} - 16 = 0

Solve the equation

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Evaluate

- 12 x^{4} - 16 = 0

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

- 12 x^{4} = 0 + 16

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

- 12 x^{4} = 16

Change the signs on both sides of the equation

12 x^{4} = - 16

Divide both sides

\frac{12 x ^{4}}{12} = \frac{- 16}{12}

Divide the numbers

x^{4} = \frac{- 16}{12}

Divide the numbers

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Evaluate

\frac{- 16}{12}

Cancel out the common factor 4

\frac{- 4}{3}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

- \frac{4}{3}

x^{4} = - \frac{4}{3}

Take the root of both sides of the equation and remember to use both positive and negative roots

x = \pm 4 - \frac{4}{3}

Simplify the expression

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Evaluate

4 - \frac{4}{3}

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

\frac{4 - 4}{4 3}

Simplify the radical expression

\frac{1 + i}{4 3}

Simplify

\frac{1}{4 3} + \frac{1}{4 3} i

Rearrange the numbers

\frac{4 27}{3} + \frac{1}{4 3} i

Rearrange the numbers

\frac{4 27}{3} + \frac{4 27}{3} i

x = \pm (\frac{4 27}{3} + \frac{4 27}{3} i)

Separate the equation into 2 possible cases

x = \frac{4 27}{3} + \frac{4 27}{3} i x = - \frac{4 27}{3} - \frac{4 27}{3} i

x = 0 x = \frac{3}{9} x = - \frac{3}{9} x = \frac{4 27}{3} + \frac{4 27}{3} i x = - \frac{4 27}{3} - \frac{4 27}{3} i

Solution

x_{1} = - \frac{4 27}{3} - \frac{4 27}{3} i, x_{2} = \frac{4 27}{3} + \frac{4 27}{3} i, x_{3} = - \frac{3}{9}, x_{4} = 0, x_{5} = \frac{3}{9}

Alternative Form

x_{1} \approx - 0.759836 - 0.759836 i, x_{2} \approx 0.759836 + 0.759836 i, x_{3} \approx - 0.19245, x_{4} = 0, x_{5} \approx 0.19245

Show Solution