Solve 5d^2(4d - 3) (2d - 1)

Question

Simplify the expression

40 d^{4} - 50 d^{3} + 15 d^{2}

Evaluate

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

Multiply the terms

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Evaluate

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

Apply the distributive property

5 d^{2} \times 4 d - 5 d^{2} \times 3

Multiply the terms

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Evaluate

5 d^{2} \times 4 d

Multiply the numbers

20 d^{2} \times d

Multiply the terms

20 d^{3}

20 d^{3} - 5 d^{2} \times 3

Multiply the numbers

20 d^{3} - 15 d^{2}

(20 d^{3} - 15 d^{2}) (2 d - 1)

Apply the distributive property

20 d^{3} \times 2 d - 20 d^{3} \times 1 - 15 d^{2} \times 2 d - (- 15 d^{2} \times 1)

Multiply the terms

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Evaluate

20 d^{3} \times 2 d

Multiply the numbers

40 d^{3} \times d

Multiply the terms

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Evaluate

d^{3} \times d

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

d^{3 + 1}

Add the numbers

d^{4}

40 d^{4}

40 d^{4} - 20 d^{3} \times 1 - 15 d^{2} \times 2 d - (- 15 d^{2} \times 1)

Any expression multiplied by 1 remains the same

40 d^{4} - 20 d^{3} - 15 d^{2} \times 2 d - (- 15 d^{2} \times 1)

Multiply the terms

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Evaluate

- 15 d^{2} \times 2 d

Multiply the numbers

- 30 d^{2} \times d

Multiply the terms

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Evaluate

d^{2} \times d

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

d^{2 + 1}

Add the numbers

d^{3}

- 30 d^{3}

40 d^{4} - 20 d^{3} - 30 d^{3} - (- 15 d^{2} \times 1)

Any expression multiplied by 1 remains the same

40 d^{4} - 20 d^{3} - 30 d^{3} - (- 15 d^{2})

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

40 d^{4} - 20 d^{3} - 30 d^{3} + 15 d^{2}

Solution

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Evaluate

- 20 d^{3} - 30 d^{3}

Collect like terms by calculating the sum or difference of their coefficients

(- 20 - 30) d^{3}

Subtract the numbers

- 50 d^{3}

40 d^{4} - 50 d^{3} + 15 d^{2}

Show Solution

Find the roots

d_{1} = 0, d_{2} = \frac{1}{2}, d_{3} = \frac{3}{4}

Alternative Form

d_{1} = 0, d_{2} = 0.5, d_{3} = 0.75

Evaluate

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

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

5 d^{2} (4 d - 3) (2 d - 1) = 0

Elimination the left coefficient

d^{2} (4 d - 3) (2 d - 1) = 0

Separate the equation into 3 possible cases

d^{2} = 0 4 d - 3 = 0 2 d - 1 = 0

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

d = 0 4 d - 3 = 0 2 d - 1 = 0

Solve the equation

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Evaluate

4 d - 3 = 0

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

4 d = 0 + 3

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

4 d = 3

Divide both sides

\frac{4 d}{4} = \frac{3}{4}

Divide the numbers

d = \frac{3}{4}

d = 0 d = \frac{3}{4} 2 d - 1 = 0

Solve the equation

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Evaluate

2 d - 1 = 0

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

2 d = 0 + 1

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

2 d = 1

Divide both sides

\frac{2 d}{2} = \frac{1}{2}

Divide the numbers

d = \frac{1}{2}

d = 0 d = \frac{3}{4} d = \frac{1}{2}

Solution

d_{1} = 0, d_{2} = \frac{1}{2}, d_{3} = \frac{3}{4}

Alternative Form

d_{1} = 0, d_{2} = 0.5, d_{3} = 0.75

Show Solution