Solve (x ^ 2 - 4x)(x ^ 2 - 4x - 1) - 20

Question

Simplify the expression

x^{4} - 8 x^{3} + 15 x^{2} + 4 x - 20

Evaluate

(x^{2} - 4 x) (x^{2} - 4 x - 1) - 20

Solution

More Steps

Evaluate

(x^{2} - 4 x) (x^{2} - 4 x - 1)

Apply the distributive property

x^{2} \times x^{2} - x^{2} \times 4 x - x^{2} \times 1 - 4 x \times x^{2} - (- 4 x \times 4 x) - (- 4 x \times 1)

Multiply the terms

More Steps

Evaluate

x^{2} \times x^{2}

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

x^{2 + 2}

Add the numbers

x^{4}

x^{4} - x^{2} \times 4 x - x^{2} \times 1 - 4 x \times x^{2} - (- 4 x \times 4 x) - (- 4 x \times 1)

Multiply the terms

More Steps

Evaluate

x^{2} \times 4 x

Use the commutative property to reorder the terms

4 x^{2} \times x

Multiply the terms

4 x^{3}

x^{4} - 4 x^{3} - x^{2} \times 1 - 4 x \times x^{2} - (- 4 x \times 4 x) - (- 4 x \times 1)

Any expression multiplied by 1 remains the same

x^{4} - 4 x^{3} - x^{2} - 4 x \times x^{2} - (- 4 x \times 4 x) - (- 4 x \times 1)

Multiply the terms

More Steps

Evaluate

x \times x^{2}

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

x^{1 + 2}

Add the numbers

x^{3}

x^{4} - 4 x^{3} - x^{2} - 4 x^{3} - (- 4 x \times 4 x) - (- 4 x \times 1)

Multiply the terms

More Steps

Evaluate

- 4 x \times 4 x

Multiply the numbers

- 16 x \times x

Multiply the terms

- 16 x^{2}

x^{4} - 4 x^{3} - x^{2} - 4 x^{3} - (- 16 x^{2}) - (- 4 x \times 1)

Any expression multiplied by 1 remains the same

x^{4} - 4 x^{3} - x^{2} - 4 x^{3} - (- 16 x^{2}) - (- 4 x)

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

x^{4} - 4 x^{3} - x^{2} - 4 x^{3} + 16 x^{2} + 4 x

Subtract the terms

More Steps

Evaluate

- 4 x^{3} - 4 x^{3}

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

(- 4 - 4) x^{3}

Subtract the numbers

- 8 x^{3}

x^{4} - 8 x^{3} - x^{2} + 16 x^{2} + 4 x

Add the terms

More Steps

Evaluate

- x^{2} + 16 x^{2}

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

(- 1 + 16) x^{2}

Add the numbers

15 x^{2}

x^{4} - 8 x^{3} + 15 x^{2} + 4 x

x^{4} - 8 x^{3} + 15 x^{2} + 4 x - 20

Show Solution

Factor the expression

(x - 5) (x - 2)^{2} (x + 1)

Evaluate

(x^{2} - 4 x) (x^{2} - 4 x - 1) - 20

Simplify

More Steps

Evaluate

(x^{2} - 4 x) (x^{2} - 4 x - 1)

Apply the distributive property

x^{2} \times x^{2} + x^{2} (- 4 x) + x^{2} (- 1) - 4 x \times x^{2} - 4 x (- 4 x) - 4 x (- 1)

Multiply the terms

More Steps

Evaluate

x^{2} \times x^{2}

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

x^{2 + 2}

Add the numbers

x^{4}

x^{4} + x^{2} (- 4 x) + x^{2} (- 1) - 4 x \times x^{2} - 4 x (- 4 x) - 4 x (- 1)

Multiply the terms

More Steps

Evaluate

x^{2} (- 4 x)

Use the commutative property to reorder the terms

- 4 x^{2} \times x

Multiply the terms

- 4 x^{3}

x^{4} - 4 x^{3} + x^{2} (- 1) - 4 x \times x^{2} - 4 x (- 4 x) - 4 x (- 1)

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

x^{4} - 4 x^{3} - x^{2} - 4 x \times x^{2} - 4 x (- 4 x) - 4 x (- 1)

Multiply the terms

More Steps

Evaluate

x \times x^{2}

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

x^{1 + 2}

Add the numbers

x^{3}

x^{4} - 4 x^{3} - x^{2} - 4 x^{3} - 4 x (- 4 x) - 4 x (- 1)

Multiply the terms

More Steps

Evaluate

- 4 x (- 4 x)

Multiply the numbers

16 x \times x

Multiply the terms

16 x^{2}

x^{4} - 4 x^{3} - x^{2} - 4 x^{3} + 16 x^{2} - 4 x (- 1)

Multiply the terms

x^{4} - 4 x^{3} - x^{2} - 4 x^{3} + 16 x^{2} + 4 x

x^{4} - 4 x^{3} - x^{2} - 4 x^{3} + 16 x^{2} + 4 x - 20

Subtract the terms

More Steps

Evaluate

- 4 x^{3} - 4 x^{3}

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

(- 4 - 4) x^{3}

Subtract the numbers

- 8 x^{3}

x^{4} - 8 x^{3} - x^{2} + 16 x^{2} + 4 x - 20

Add the terms

More Steps

Evaluate

- x^{2} + 16 x^{2}

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

(- 1 + 16) x^{2}

Add the numbers

15 x^{2}

x^{4} - 8 x^{3} + 15 x^{2} + 4 x - 20

Evaluate

x^{4} - 8 x^{3} + 4 x + 15 x^{2} - 20

Calculate

x^{4} - 3 x^{3} + 4 x - 5 x^{3} + 15 x^{2} - 20

Rewrite the expression

x \times x^{3} - x \times 3 x^{2} + x \times 4 - 5 x^{3} + 5 \times 3 x^{2} - 5 \times 4

Factor out x from the expression

x (x^{3} - 3 x^{2} + 4) - 5 x^{3} + 5 \times 3 x^{2} - 5 \times 4

Factor out - 5 from the expression

x (x^{3} - 3 x^{2} + 4) - 5 (x^{3} - 3 x^{2} + 4)

Factor out x^{3} - 3 x^{2} + 4 from the expression

(x - 5) (x^{3} - 3 x^{2} + 4)

Factor the expression

More Steps

Evaluate

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

Calculate

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

Rewrite the expression

x^{2} \times x + x^{2} - 4 x \times x - 4 x + 4 x + 4

Factor out x^{2} from the expression

x^{2} (x + 1) - 4 x \times x - 4 x + 4 x + 4

Factor out - 4 x from the expression

x^{2} (x + 1) - 4 x (x + 1) + 4 x + 4

Factor out 4 from the expression

x^{2} (x + 1) - 4 x (x + 1) + 4 (x + 1)

Factor out x + 1 from the expression

(x^{2} - 4 x + 4) (x + 1)

(x - 5) (x^{2} - 4 x + 4) (x + 1)

Solution

(x - 5) (x - 2)^{2} (x + 1)

Show Solution

Find the roots

x_{1} = - 1, x_{2} = 2, x_{3} = 5

Evaluate

(x^{2} - 4 x) (x^{2} - 4 x - 1) - 20

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

(x^{2} - 4 x) (x^{2} - 4 x - 1) - 20 = 0

Calculate

More Steps

Evaluate

(x^{2} - 4 x) (x^{2} - 4 x - 1)

Apply the distributive property

x^{2} \times x^{2} - x^{2} \times 4 x - x^{2} \times 1 - 4 x \times x^{2} - (- 4 x \times 4 x) - (- 4 x \times 1)

Multiply the terms

More Steps

Evaluate

x^{2} \times x^{2}

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

x^{2 + 2}

Add the numbers

x^{4}

x^{4} - x^{2} \times 4 x - x^{2} \times 1 - 4 x \times x^{2} - (- 4 x \times 4 x) - (- 4 x \times 1)

Multiply the terms

More Steps

Evaluate

x^{2} \times 4 x

Use the commutative property to reorder the terms

4 x^{2} \times x

Multiply the terms

4 x^{3}

x^{4} - 4 x^{3} - x^{2} \times 1 - 4 x \times x^{2} - (- 4 x \times 4 x) - (- 4 x \times 1)

Any expression multiplied by 1 remains the same

x^{4} - 4 x^{3} - x^{2} - 4 x \times x^{2} - (- 4 x \times 4 x) - (- 4 x \times 1)

Multiply the terms

More Steps

Evaluate

x \times x^{2}

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

x^{1 + 2}

Add the numbers

x^{3}

x^{4} - 4 x^{3} - x^{2} - 4 x^{3} - (- 4 x \times 4 x) - (- 4 x \times 1)

Multiply the terms

More Steps

Evaluate

- 4 x \times 4 x

Multiply the numbers

- 16 x \times x

Multiply the terms

- 16 x^{2}

x^{4} - 4 x^{3} - x^{2} - 4 x^{3} - (- 16 x^{2}) - (- 4 x \times 1)

Any expression multiplied by 1 remains the same

x^{4} - 4 x^{3} - x^{2} - 4 x^{3} - (- 16 x^{2}) - (- 4 x)

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

x^{4} - 4 x^{3} - x^{2} - 4 x^{3} + 16 x^{2} + 4 x

Subtract the terms

More Steps

Evaluate

- 4 x^{3} - 4 x^{3}

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

(- 4 - 4) x^{3}

Subtract the numbers

- 8 x^{3}

x^{4} - 8 x^{3} - x^{2} + 16 x^{2} + 4 x

Add the terms

More Steps

Evaluate

- x^{2} + 16 x^{2}

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

(- 1 + 16) x^{2}

Add the numbers

15 x^{2}

x^{4} - 8 x^{3} + 15 x^{2} + 4 x

x^{4} - 8 x^{3} + 15 x^{2} + 4 x - 20 = 0

Factor the expression

(x - 5) (x - 2)^{2} (x + 1) = 0

Separate the equation into 3 possible cases

x - 5 = 0 (x - 2)^{2} = 0 x + 1 = 0

Solve the equation

More Steps

Evaluate

x - 5 = 0

Move the constant to the right side

x = 0 + 5

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

x = 5

x = 5 (x - 2)^{2} = 0 x + 1 = 0

Solve the equation

More Steps

Evaluate

(x - 2)^{2} = 0

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

x - 2 = 0

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

x = 0 + 2

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

x = 2

x = 5 x = 2 x + 1 = 0

Solve the equation

More Steps

Evaluate

x + 1 = 0

Move the constant to the right side

x = 0 - 1

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

x = - 1

x = 5 x = 2 x = - 1

Solution

x_{1} = - 1, x_{2} = 2, x_{3} = 5

Show Solution