Solve (x^2-x-1)(x^2-x-7)

Question

Simplify the expression

x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 7

Evaluate

(x^{2} - x - 1) (x^{2} - x - 7)

Apply the distributive property

x^{2} \times x^{2} - x^{2} \times x - x^{2} \times 7 - x \times x^{2} - (- x \times x) - (- x \times 7) - x^{2} - (- x) - (- 7)

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 x - x^{2} \times 7 - x \times x^{2} - (- x \times x) - (- x \times 7) - x^{2} - (- x) - (- 7)

Multiply the terms

More Steps

Evaluate

x^{2} \times x

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

x^{2 + 1}

Add the numbers

x^{3}

x^{4} - x^{3} - x^{2} \times 7 - x \times x^{2} - (- x \times x) - (- x \times 7) - x^{2} - (- x) - (- 7)

Use the commutative property to reorder the terms

x^{4} - x^{3} - 7 x^{2} - x \times x^{2} - (- x \times x) - (- x \times 7) - x^{2} - (- x) - (- 7)

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} - x^{3} - 7 x^{2} - x^{3} - (- x \times x) - (- x \times 7) - x^{2} - (- x) - (- 7)

Multiply the terms

x^{4} - x^{3} - 7 x^{2} - x^{3} - (- x^{2}) - (- x \times 7) - x^{2} - (- x) - (- 7)

Use the commutative property to reorder the terms

x^{4} - x^{3} - 7 x^{2} - x^{3} - (- x^{2}) - (- 7 x) - x^{2} - (- x) - (- 7)

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} - x^{3} - 7 x^{2} - x^{3} + x^{2} + 7 x - x^{2} + x + 7

Subtract the terms

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Evaluate

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

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

(- 1 - 1) x^{3}

Subtract the numbers

- 2 x^{3}

x^{4} - 2 x^{3} - 7 x^{2} + x^{2} + 7 x - x^{2} + x + 7

Calculate the sum or difference

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Evaluate

- 7 x^{2} + x^{2} - x^{2}

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

(- 7 + 1 - 1) x^{2}

Calculate the sum or difference

- 7 x^{2}

x^{4} - 2 x^{3} - 7 x^{2} + 7 x + x + 7

Solution

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Evaluate

7 x + x

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

(7 + 1) x

Add the numbers

8 x

x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 7

Show Solution

Find the roots

x_{1} = \frac{1 - 29}{2}, x_{2} = \frac{1 - 5}{2}, x_{3} = \frac{1 + 5}{2}, x_{4} = \frac{1 + 29}{2}

Alternative Form

x_{1} \approx - 2.192582, x_{2} \approx - 0.618034, x_{3} \approx 1.618034, x_{4} \approx 3.192582

Evaluate

(x^{2} - x - 1) (x^{2} - x - 7)

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

(x^{2} - x - 1) (x^{2} - x - 7) = 0

Separate the equation into 2 possible cases

x^{2} - x - 1 = 0 x^{2} - x - 7 = 0

Solve the equation

More Steps

Evaluate

x^{2} - x - 1 = 0

Substitute a= 1,b= - 1 and c= - 1 into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{1 \pm ( - 1 ) ^{2} - 4 ( - 1 )}{2}

Simplify the expression

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Evaluate

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

Evaluate the power

1 - 4 (- 1)

Simplify

1 - (- 4)

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

1 + 4

Add the numbers

5

x = \frac{1 \pm 5}{2}

Separate the equation into 2 possible cases

x = \frac{1 + 5}{2} x = \frac{1 - 5}{2}

x = \frac{1 + 5}{2} x = \frac{1 - 5}{2} x^{2} - x - 7 = 0

Solve the equation

More Steps

Evaluate

x^{2} - x - 7 = 0

Substitute a= 1,b= - 1 and c= - 7 into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{1 \pm ( - 1 ) ^{2} - 4 ( - 7 )}{2}

Simplify the expression

More Steps

Evaluate

(- 1)^{2} - 4 (- 7)

Evaluate the power

1 - 4 (- 7)

Multiply the numbers

1 - (- 28)

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

1 + 28

Add the numbers

29

x = \frac{1 \pm 29}{2}

Separate the equation into 2 possible cases

x = \frac{1 + 29}{2} x = \frac{1 - 29}{2}

x = \frac{1 + 5}{2} x = \frac{1 - 5}{2} x = \frac{1 + 29}{2} x = \frac{1 - 29}{2}

Solution

x_{1} = \frac{1 - 29}{2}, x_{2} = \frac{1 - 5}{2}, x_{3} = \frac{1 + 5}{2}, x_{4} = \frac{1 + 29}{2}

Alternative Form

x_{1} \approx - 2.192582, x_{2} \approx - 0.618034, x_{3} \approx 1.618034, x_{4} \approx 3.192582

Show Solution