Solve -x^{2}-\frac{1}{x^{2}-1} - Socratic AI (ScanMath)

Pregunta

Simplify the expression

\frac{- x ^{4} + x ^{2} - 1}{x ^{2} - 1}

Evaluate

- x^{2} - \frac{1}{x ^{2} - 1}

Reduce fractions to a common denominator

- \frac{x ^{2} ( x ^{2} - 1 )}{x ^{2} - 1} - \frac{1}{x ^{2} - 1}

Write all numerators above the common denominator

\frac{- x ^{2} ( x ^{2} - 1 ) - 1}{x ^{2} - 1}

Multiply the terms

Más Pasos

Evaluate

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

Apply the distributive property

x^{2} \times x^{2} - x^{2} \times 1

Multiply the terms

Más Pasos

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 1

Any expression multiplied by 1 remains the same

x^{4} - x^{2}

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

Solución

\frac{- x ^{4} + x ^{2} - 1}{x ^{2} - 1}

Mostrar solución

Find the excluded values

x = 1, x = - 1

Evaluate

- x^{2} - \frac{1}{x ^{2} - 1}

To find the excluded values,set the denominators equal to 0

x^{2} - 1 = 0

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

x^{2} = 0 + 1

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

x^{2} = 1

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

x = \pm 1

Simplify the expression

x = \pm 1

Separate the equation into 2 possible cases

x = 1 x = - 1

Solución

x = 1, x = - 1

Mostrar solución

Find the roots

x \in / R

Evaluate

- x^{2} - \frac{1}{x ^{2} - 1}

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

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

Find the domain

Más Pasos

Evaluate

x^{2} - 1 \neq = 0

Move the constant to the right side

x^{2} \neq = 1

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

x \neq = \pm 1

Simplify the expression

x \neq = \pm 1

Separate the inequality into 2 possible cases

{x \neq = 1 x \neq = - 1

Find the intersection

x \in (- \infty, - 1) \cup (- 1, 1) \cup (1, + \infty)

- x^{2} - \frac{1}{x ^{2} - 1} = 0, x \in (- \infty, - 1) \cup (- 1, 1) \cup (1, + \infty)

Calculate

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

Subtract the terms

Más Pasos

Simplify

- x^{2} - \frac{1}{x ^{2} - 1}

Reduce fractions to a common denominator

- \frac{x ^{2} ( x ^{2} - 1 )}{x ^{2} - 1} - \frac{1}{x ^{2} - 1}

Write all numerators above the common denominator

\frac{- x ^{2} ( x ^{2} - 1 ) - 1}{x ^{2} - 1}

Multiply the terms

Más Pasos

Evaluate

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

Apply the distributive property

x^{2} \times x^{2} - x^{2} \times 1

Multiply the terms

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

Any expression multiplied by 1 remains the same

x^{4} - x^{2}

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

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

\frac{- x ^{4} + x ^{2} - 1}{x ^{2} - 1}

\frac{- x ^{4} + x ^{2} - 1}{x ^{2} - 1} = 0

Cross multiply

- x^{4} + x^{2} - 1 = (x^{2} - 1) \times 0

Simplify the equation

- x^{4} + x^{2} - 1 = 0

Solve the equation using substitution t = x^{2}

- t^{2} + t - 1 = 0

Multiply both sides

t^{2} - t + 1 = 0

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

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

Simplify the expression

Más Pasos

Evaluate

(- 1)^{2} - 4

Evaluate the power

1 - 4

Subtract the numbers

- 3

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

The expression is undefined in the set of real numbers

t \in / R

Solución

x \in / R

Mostrar solución

X^{2} \frac{1}{x^{2} 1}

Simplify the expression

Find the excluded values

Find the roots