Solve (-2t^2 3t-1)(2t^2 5t)

Question

Simplify the expression

- 60 t^{6} - 10 t^{3}

Evaluate

(- 2 t^{2} \times 3 t - 1) (2 t^{2} \times 5 t)

Remove the parentheses

(- 2 t^{2} \times 3 t - 1) \times 2 t^{2} \times 5 t

Multiply

More Steps

Multiply the terms

- 2 t^{2} \times 3 t

Multiply the terms

- 6 t^{2} \times t

Multiply the terms with the same base by adding their exponents

- 6 t^{2 + 1}

Add the numbers

- 6 t^{3}

(- 6 t^{3} - 1) \times 2 t^{2} \times 5 t

Multiply the terms

(- 6 t^{3} - 1) \times 10 t^{2} \times t

Multiply the terms with the same base by adding their exponents

(- 6 t^{3} - 1) \times 10 t^{2 + 1}

Add the numbers

(- 6 t^{3} - 1) \times 10 t^{3}

Multiply the terms

10 t^{3} (- 6 t^{3} - 1)

Apply the distributive property

10 t^{3} (- 6 t^{3}) - 10 t^{3} \times 1

Multiply the terms

More Steps

Evaluate

10 t^{3} (- 6 t^{3})

Multiply the numbers

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Evaluate

10 (- 6)

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

- 10 \times 6

Multiply the numbers

- 60

- 60 t^{3} \times t^{3}

Multiply the terms

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Evaluate

t^{3} \times t^{3}

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

t^{3 + 3}

Add the numbers

t^{6}

- 60 t^{6}

- 60 t^{6} - 10 t^{3} \times 1

Solution

- 60 t^{6} - 10 t^{3}

Show Solution

Find the roots

t_{1} = - \frac{3 36}{6}, t_{2} = 0

Alternative Form

t_{1} \approx - 0.550321, t_{2} = 0

Evaluate

(- 2 t^{2} \times 3 t - 1) (2 t^{2} \times 5 t)

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

(- 2 t^{2} \times 3 t - 1) (2 t^{2} \times 5 t) = 0

Multiply

More Steps

Multiply the terms

- 2 t^{2} \times 3 t

Multiply the terms

- 6 t^{2} \times t

Multiply the terms with the same base by adding their exponents

- 6 t^{2 + 1}

Add the numbers

- 6 t^{3}

(- 6 t^{3} - 1) (2 t^{2} \times 5 t) = 0

Multiply

More Steps

Multiply the terms

2 t^{2} \times 5 t

Multiply the terms

10 t^{2} \times t

Multiply the terms with the same base by adding their exponents

10 t^{2 + 1}

Add the numbers

10 t^{3}

(- 6 t^{3} - 1) \times 10 t^{3} = 0

Multiply the terms

10 t^{3} (- 6 t^{3} - 1) = 0

Elimination the left coefficient

t^{3} (- 6 t^{3} - 1) = 0

Separate the equation into 2 possible cases

t^{3} = 0 - 6 t^{3} - 1 = 0

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

t = 0 - 6 t^{3} - 1 = 0

Solve the equation

More Steps

Evaluate

- 6 t^{3} - 1 = 0

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

- 6 t^{3} = 0 + 1

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

- 6 t^{3} = 1

Change the signs on both sides of the equation

6 t^{3} = - 1

Divide both sides

\frac{6 t ^{3}}{6} = \frac{- 1}{6}

Divide the numbers

t^{3} = \frac{- 1}{6}

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

t^{3} = - \frac{1}{6}

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

3 t^{3} = 3 - \frac{1}{6}

Calculate

t = 3 - \frac{1}{6}

Simplify the root

More Steps

Evaluate

3 - \frac{1}{6}

An odd root of a negative radicand is always a negative

- 3 \frac{1}{6}

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

- \frac{3 1}{3 6}

Simplify the radical expression

- \frac{1}{3 6}

Multiply by the Conjugate

\frac{- 3 6 ^{2}}{3 6 \times 3 6 ^{2}}

Simplify

\frac{- 3 36}{3 6 \times 3 6 ^{2}}

Multiply the numbers

\frac{- 3 36}{6}

Calculate

- \frac{3 36}{6}

t = - \frac{3 36}{6}

t = 0 t = - \frac{3 36}{6}

Solution

t_{1} = - \frac{3 36}{6}, t_{2} = 0

Alternative Form

t_{1} \approx - 0.550321, t_{2} = 0

Show Solution