Solve -t^3 4t^2 9t - 36

Question

Simplify the expression

- 36 t^{6} - 36

Evaluate

- t^{3} \times 4 t^{2} \times 9 t - 36

Solution

More Steps

Evaluate

- t^{3} \times 4 t^{2} \times 9 t

Multiply the terms with the same base by adding their exponents

- t^{3 + 2 + 1} \times 4 \times 9

Add the numbers

- t^{6} \times 4 \times 9

Multiply the terms

- t^{6} \times 36

Use the commutative property to reorder the terms

- 36 t^{6}

- 36 t^{6} - 36

Show Solution

Factor the expression

- 36 (t^{2} + 1) (t^{4} - t^{2} + 1)

Evaluate

- t^{3} \times 4 t^{2} \times 9 t - 36

Evaluate

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Evaluate

t^{3} \times 4 t^{2} \times 9 t

Multiply the terms with the same base by adding their exponents

t^{3 + 2 + 1} \times 4 \times 9

Add the numbers

t^{6} \times 4 \times 9

Multiply the terms

t^{6} \times 36

Use the commutative property to reorder the terms

36 t^{6}

- 36 t^{6} - 36

Factor out - 36 from the expression

- 36 (t^{6} + 1)

Solution

More Steps

Evaluate

t^{6} + 1

Rewrite the expression in exponential form

t^{6} + 1^{3}

Use a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2}) to factor the expression

(t^{2} + 1) (t^{4} - t^{2} \times 1 + 1^{2})

Any expression multiplied by 1 remains the same

(t^{2} + 1) (t^{4} - t^{2} + 1^{2})

1 raised to any power equals to 1

(t^{2} + 1) (t^{4} - t^{2} + 1)

- 36 (t^{2} + 1) (t^{4} - t^{2} + 1)

Show Solution

Find the roots

t_{1} = - \frac{3}{2} - \frac{1}{2} i, t_{2} = \frac{3}{2} + \frac{1}{2} i

Alternative Form

t_{1} \approx - 0.866025 - 0.5 i, t_{2} \approx 0.866025 + 0.5 i

Evaluate

- t^{3} \times 4 t^{2} \times 9 t - 36

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

- t^{3} \times 4 t^{2} \times 9 t - 36 = 0

Multiply

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Multiply the terms

t^{3} \times 4 t^{2} \times 9 t

Multiply the terms with the same base by adding their exponents

t^{3 + 2 + 1} \times 4 \times 9

Add the numbers

t^{6} \times 4 \times 9

Multiply the terms

t^{6} \times 36

Use the commutative property to reorder the terms

36 t^{6}

- 36 t^{6} - 36 = 0

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

- 36 t^{6} = 0 + 36

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

- 36 t^{6} = 36

Change the signs on both sides of the equation

36 t^{6} = - 36

Divide both sides

\frac{36 t ^{6}}{36} = \frac{- 36}{36}

Divide the numbers

t^{6} = \frac{- 36}{36}

Divide the numbers

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Evaluate

\frac{- 36}{36}

Reduce the numbers

\frac{- 1}{1}

Calculate

- 1

t^{6} = - 1

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

t = \pm 6 - 1

Simplify the expression

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Evaluate

6 - 1

Rewrite the expression

1 \times (\frac{3}{2} + \frac{1}{2} i)

Any expression multiplied by 1 remains the same

\frac{3}{2} + \frac{1}{2} i

t = \pm (\frac{3}{2} + \frac{1}{2} i)

Separate the equation into 2 possible cases

t = \frac{3}{2} + \frac{1}{2} i t = - \frac{3}{2} - \frac{1}{2} i

Solution

t_{1} = - \frac{3}{2} - \frac{1}{2} i, t_{2} = \frac{3}{2} + \frac{1}{2} i

Alternative Form

t_{1} \approx - 0.866025 - 0.5 i, t_{2} \approx 0.866025 + 0.5 i

Show Solution