Solve x^4-6x^2-7x-6

Question

Factor the expression

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

Evaluate

x^{4} - 6 x^{2} - 7 x - 6

Calculate

x^{4} + 3 x^{3} + 3 x^{2} + 2 x - 3 x^{3} - 9 x^{2} - 9 x - 6

Rewrite the expression

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

Factor out x from the expression

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

Factor out - 3 from the expression

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

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

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

Solution

More Steps

Evaluate

x^{3} + 3 x^{2} + 3 x + 2

Calculate

x^{3} + x^{2} + x + 2 x^{2} + 2 x + 2

Rewrite the expression

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

Factor out x from the expression

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

Factor out 2 from the expression

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

Factor out x^{2} + x + 1 from the expression

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

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

Show Solution

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

Alternative Form

x_{1} \approx - 0.5 - 0.866025 i, x_{2} \approx - 0.5 + 0.866025 i, x_{3} = - 2, x_{4} = 3

Evaluate

x^{4} - 6 x^{2} - 7 x - 6

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

x^{4} - 6 x^{2} - 7 x - 6 = 0

Factor the expression

(x - 3) (x + 2) (x^{2} + x + 1) = 0

Separate the equation into 3 possible cases

x - 3 = 0 x + 2 = 0 x^{2} + x + 1 = 0

Solve the equation

More Steps

Evaluate

x - 3 = 0

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

x = 0 + 3

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

x = 3

x = 3 x + 2 = 0 x^{2} + x + 1 = 0

Solve the equation

More Steps

Evaluate

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 = 3 x = - 2 x^{2} + x + 1 = 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}{2}

Simplify the expression

More Steps

Evaluate

1^{2} - 4

1 raised to any power equals to 1

1 - 4

Subtract the numbers

- 3

x = \frac{- 1 \pm - 3}{2}

Simplify the radical expression

More Steps

Evaluate

- 3

Evaluate the power

3 \times - 1

Evaluate the power

3 \times i

x = \frac{- 1 \pm 3 \times i}{2}

Separate the equation into 2 possible cases

x = \frac{- 1 + 3 \times i}{2} x = \frac{- 1 - 3 \times i}{2}

Simplify the expression

x = - \frac{1}{2} + \frac{3}{2} i x = \frac{- 1 - 3 \times i}{2}

Simplify the expression

x = - \frac{1}{2} + \frac{3}{2} i x = - \frac{1}{2} - \frac{3}{2} i

x = 3 x = - 2 x = - \frac{1}{2} + \frac{3}{2} i x = - \frac{1}{2} - \frac{3}{2} i

Solution

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

Alternative Form

x_{1} \approx - 0.5 - 0.866025 i, x_{2} \approx - 0.5 + 0.866025 i, x_{3} = - 2, x_{4} = 3

Show Solution