Solve (5p^2-3) (2p^2-3p^3)

Question

Simplify the expression

10 p^{4} - 15 p^{5} - 6 p^{2} + 9 p^{3}

Evaluate

(5 p^{2} - 3) (2 p^{2} - 3 p^{3})

Apply the distributive property

5 p^{2} \times 2 p^{2} - 5 p^{2} \times 3 p^{3} - 3 \times 2 p^{2} - (- 3 \times 3 p^{3})

Multiply the terms

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Evaluate

5 p^{2} \times 2 p^{2}

Multiply the numbers

10 p^{2} \times p^{2}

Multiply the terms

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Evaluate

p^{2} \times p^{2}

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

p^{2 + 2}

Add the numbers

p^{4}

10 p^{4}

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

Multiply the terms

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Evaluate

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

Multiply the numbers

15 p^{2} \times p^{3}

Multiply the terms

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Evaluate

p^{2} \times p^{3}

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

p^{2 + 3}

Add the numbers

p^{5}

15 p^{5}

10 p^{4} - 15 p^{5} - 3 \times 2 p^{2} - (- 3 \times 3 p^{3})

Multiply the numbers

10 p^{4} - 15 p^{5} - 6 p^{2} - (- 3 \times 3 p^{3})

Multiply the numbers

10 p^{4} - 15 p^{5} - 6 p^{2} - (- 9 p^{3})

Solution

10 p^{4} - 15 p^{5} - 6 p^{2} + 9 p^{3}

Show Solution

Factor the expression

p^{2} (5 p^{2} - 3) (2 - 3 p)

Evaluate

(5 p^{2} - 3) (2 p^{2} - 3 p^{3})

Factor the expression

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Evaluate

2 p^{2} - 3 p^{3}

Rewrite the expression

p^{2} \times 2 - p^{2} \times 3 p

Factor out p^{2} from the expression

p^{2} (2 - 3 p)

(5 p^{2} - 3) p^{2} (2 - 3 p)

Solution

p^{2} (5 p^{2} - 3) (2 - 3 p)

Show Solution

Find the roots

p_{1} = - \frac{15}{5}, p_{2} = 0, p_{3} = \frac{2}{3}, p_{4} = \frac{15}{5}

Alternative Form

p_{1} \approx - 0.774597, p_{2} = 0, p_{3} = 0. \dot{6}, p_{4} \approx 0.774597

Evaluate

(5 p^{2} - 3) (2 p^{2} - 3 p^{3})

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

(5 p^{2} - 3) (2 p^{2} - 3 p^{3}) = 0

Separate the equation into 2 possible cases

5 p^{2} - 3 = 0 2 p^{2} - 3 p^{3} = 0

Solve the equation

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Evaluate

5 p^{2} - 3 = 0

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

5 p^{2} = 0 + 3

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

5 p^{2} = 3

Divide both sides

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

Divide the numbers

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

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

p = \pm \frac{3}{5}

Simplify the expression

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Evaluate

\frac{3}{5}

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

\frac{3}{5}

Multiply by the Conjugate

\frac{3 \times 5}{5 \times 5}

Multiply the numbers

\frac{15}{5 \times 5}

When a square root of an expression is multiplied by itself,the result is that expression

\frac{15}{5}

p = \pm \frac{15}{5}

Separate the equation into 2 possible cases

p = \frac{15}{5} p = - \frac{15}{5}

p = \frac{15}{5} p = - \frac{15}{5} 2 p^{2} - 3 p^{3} = 0

Solve the equation

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Evaluate

2 p^{2} - 3 p^{3} = 0

Factor the expression

p^{2} (2 - 3 p) = 0

Separate the equation into 2 possible cases

p^{2} = 0 2 - 3 p = 0

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

p = 0 2 - 3 p = 0

Solve the equation

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Evaluate

2 - 3 p = 0

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

- 3 p = 0 - 2

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

- 3 p = - 2

Change the signs on both sides of the equation

3 p = 2

Divide both sides

\frac{3 p}{3} = \frac{2}{3}

Divide the numbers

p = \frac{2}{3}

p = 0 p = \frac{2}{3}

p = \frac{15}{5} p = - \frac{15}{5} p = 0 p = \frac{2}{3}

Solution

p_{1} = - \frac{15}{5}, p_{2} = 0, p_{3} = \frac{2}{3}, p_{4} = \frac{15}{5}

Alternative Form

p_{1} \approx - 0.774597, p_{2} = 0, p_{3} = 0. \dot{6}, p_{4} \approx 0.774597

Show Solution