Solve (1-a^2)(2a-a^2)

Question

Simplify the expression

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

Evaluate

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

Apply the distributive property

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

Any expression multiplied by 1 remains the same

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

Any expression multiplied by 1 remains the same

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

Multiply the terms

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Evaluate

- a^{2} \times 2 a

Multiply the numbers

- 2 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}

- 2 a^{3}

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

Multiply the terms

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Evaluate

a^{2} \times a^{2}

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

a^{2 + 2}

Add the numbers

a^{4}

2 a - a^{2} - 2 a^{3} - (- a^{4})

Solution

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

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Factor the expression

a (1 - a) (1 + a) (2 - a)

Evaluate

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

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

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

Factor the expression

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Evaluate

2 a - a^{2}

Rewrite the expression

a \times 2 - a \times a

Factor out a from the expression

a (2 - a)

(1 - a) (1 + a) a (2 - a)

Solution

a (1 - a) (1 + a) (2 - a)

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Find the roots

a_{1} = - 1, a_{2} = 0, a_{3} = 1, a_{4} = 2

Evaluate

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

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

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

Separate the equation into 2 possible cases

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

Solve the equation

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Evaluate

1 - a^{2} = 0

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

- a^{2} = 0 - 1

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

- a^{2} = - 1

Change the signs on both sides of the equation

a^{2} = 1

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

a = \pm 1

Simplify the expression

a = \pm 1

Separate the equation into 2 possible cases

a = 1 a = - 1

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

Solve the equation

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Evaluate

2 a - a^{2} = 0

Factor the expression

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Evaluate

2 a - a^{2}

Rewrite the expression

a \times 2 - a \times a

Factor out a from the expression

a (2 - a)

a (2 - a) = 0

When the product of factors equals 0,at least one factor is 0

a = 0 2 - a = 0

Solve the equation for a

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Evaluate

2 - a = 0

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

- a = 0 - 2

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

- a = - 2

Change the signs on both sides of the equation

a = 2

a = 0 a = 2

a = 1 a = - 1 a = 0 a = 2

Solution

a_{1} = - 1, a_{2} = 0, a_{3} = 1, a_{4} = 2

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