Solve -10a(a-1) 2a(3-a)

Question

Simplify the expression

- 80 a^{3} + 20 a^{4} + 60 a^{2}

Evaluate

- 10 a (a - 1) \times 2 a (3 - a)

Multiply the terms

- 20 a (a - 1) a (3 - a)

Multiply the terms

- 20 a^{2} (a - 1) (3 - a)

Multiply the terms

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Evaluate

- 20 a^{2} (a - 1)

Apply the distributive property

- 20 a^{2} \times a - (- 20 a^{2} \times 1)

Multiply the terms

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Evaluate

a^{2} \times a

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

a^{2 + 1}

Add the numbers

a^{3}

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

Any expression multiplied by 1 remains the same

- 20 a^{3} - (- 20 a^{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

- 20 a^{3} + 20 a^{2}

(- 20 a^{3} + 20 a^{2}) (3 - a)

Apply the distributive property

- 20 a^{3} \times 3 - (- 20 a^{3} \times a) + 20 a^{2} \times 3 - 20 a^{2} \times a

Multiply the numbers

- 60 a^{3} - (- 20 a^{3} \times a) + 20 a^{2} \times 3 - 20 a^{2} \times a

Multiply the terms

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Evaluate

a^{3} \times a

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

a^{3 + 1}

Add the numbers

a^{4}

- 60 a^{3} - (- 20 a^{4}) + 20 a^{2} \times 3 - 20 a^{2} \times a

Multiply the numbers

- 60 a^{3} - (- 20 a^{4}) + 60 a^{2} - 20 a^{2} \times a

Multiply the terms

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Evaluate

a^{2} \times a

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

a^{2 + 1}

Add the numbers

a^{3}

- 60 a^{3} - (- 20 a^{4}) + 60 a^{2} - 20 a^{3}

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

- 60 a^{3} + 20 a^{4} + 60 a^{2} - 20 a^{3}

Solution

More Steps

Evaluate

- 60 a^{3} - 20 a^{3}

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

(- 60 - 20) a^{3}

Subtract the numbers

- 80 a^{3}

- 80 a^{3} + 20 a^{4} + 60 a^{2}

Show Solution

Find the roots

a_{1} = 0, a_{2} = 1, a_{3} = 3

Evaluate

- 10 a (a - 1) \times 2 a (3 - a)

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

- 10 a (a - 1) \times 2 a (3 - a) = 0

Multiply

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Multiply the terms

- 10 a (a - 1) \times 2 a (3 - a)

Multiply the terms

- 20 a (a - 1) a (3 - a)

Multiply the terms

- 20 a^{2} (a - 1) (3 - a)

- 20 a^{2} (a - 1) (3 - a) = 0

Change the sign

20 a^{2} (a - 1) (3 - a) = 0

Elimination the left coefficient

a^{2} (a - 1) (3 - a) = 0

Separate the equation into 3 possible cases

a^{2} = 0 a - 1 = 0 3 - a = 0

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

a = 0 a - 1 = 0 3 - a = 0

Solve the equation

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Evaluate

a - 1 = 0

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

a = 0 + 1

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

a = 1

a = 0 a = 1 3 - a = 0

Solve the equation

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Evaluate

3 - a = 0

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

- a = 0 - 3

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

- a = - 3

Change the signs on both sides of the equation

a = 3

a = 0 a = 1 a = 3

Solution

a_{1} = 0, a_{2} = 1, a_{3} = 3

Show Solution