Solve 3x^(2)-x^5x-2

Question

Simplify the expression

3 x^{2} - x^{6} - 2

Evaluate

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

Solution

More Steps

Evaluate

x^{5} \times x

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

x^{5 + 1}

Add the numbers

x^{6}

3 x^{2} - x^{6} - 2

Show Solution

Factor the expression

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

Evaluate

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

Multiply the terms

More Steps

Evaluate

x^{5} \times x

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

x^{5 + 1}

Add the numbers

x^{6}

3 x^{2} - x^{6} - 2

Evaluate

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

Calculate

- x^{6} - 3 x^{4} - 2 x^{5} - 4 x^{3} - 2 x^{2} + 2 x^{5} + 6 x^{3} + 4 x^{4} + 8 x^{2} + 4 x - x^{4} - 3 x^{2} - 2 x^{3} - 4 x - 2

Rewrite the expression

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

Factor out x^{2} from the expression

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

Factor out - 2 x from the expression

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

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

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

Factor the expression

More Steps

Evaluate

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

Calculate

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

Rewrite the expression

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

Factor out x^{2} from the expression

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

Factor out 2 x from the expression

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

Factor out - x^{2} - 2 from the expression

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

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

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

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

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

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

Solution

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

Show Solution

Find the roots

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

Alternative Form

x_{1} \approx - 1.414214 i, x_{2} \approx 1.414214 i, x_{3} = - 1, x_{4} = 1

Evaluate

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

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

3 x^{2} - x^{5} \times x - 2 = 0

Multiply the terms

More Steps

Evaluate

x^{5} \times x

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

x^{5 + 1}

Add the numbers

x^{6}

3 x^{2} - x^{6} - 2 = 0

Factor the expression

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

Separate the equation into 3 possible cases

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

Solve the equation

More Steps

Evaluate

(x - 1)^{2} = 0

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

x - 1 = 0

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

x = 0 + 1

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

x = 1

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

Solve the equation

More Steps

Evaluate

(x + 1)^{2} = 0

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

x + 1 = 0

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

x = 0 - 1

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

x = - 1

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

Solve the equation

More Steps

Evaluate

- x^{2} - 2 = 0

Move the constant to the right side

- x^{2} = 2

Change the signs on both sides of the equation

x^{2} = - 2

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

x = \pm - 2

Simplify the expression

More Steps

Evaluate

- 2

Evaluate the power

2 \times - 1

Evaluate the power

2 \times i

x = \pm (2 \times i)

Separate the equation into 2 possible cases

x = 2 \times i x = - 2 \times i

x = 1 x = - 1 x = 2 \times i x = - 2 \times i

Solution

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

Alternative Form

x_{1} \approx - 1.414214 i, x_{2} \approx 1.414214 i, x_{3} = - 1, x_{4} = 1

Show Solution