Solve 4x^4-x^2-4<=0

Question

Solve the inequality

- \frac{2 65 + 2}{4} \leq x \leq \frac{2 65 + 2}{4}

Alternative Form

x \in - \frac{2 65 + 2}{4}, \frac{2 65 + 2}{4}

Evaluate

4 x^{4} - x^{2} - 4 \leq 0

Rewrite the expression

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

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

4 t^{2} - t - 4 = 0

Add or subtract both sides

4 t^{2} - t = 4

Divide both sides

\frac{4 t ^{2} - t}{4} = \frac{4}{4}

Evaluate

t^{2} - \frac{1}{4} t = 1

Add the same value to both sides

t^{2} - \frac{1}{4} t + \frac{1}{64} = 1 + \frac{1}{64}

Simplify the expression

(t - \frac{1}{8})^{2} = \frac{65}{64}

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

t - \frac{1}{8} = \pm \frac{65}{64}

Simplify the expression

t - \frac{1}{8} = \pm \frac{65}{8}

Separate the equation into 2 possible cases

t - \frac{1}{8} = \frac{65}{8} t - \frac{1}{8} = - \frac{65}{8}

Solve the equation

More Steps

t = \frac{65 + 1}{8} t - \frac{1}{8} = - \frac{65}{8}

Solve the equation

More Steps

t = \frac{65 + 1}{8} t = \frac{- 65 + 1}{8}

Substitute back

x^{2} = \frac{65 + 1}{8} x^{2} = \frac{- 65 + 1}{8}

Solve the equation for x

More Steps

x = \frac{2 65 + 2}{4} x = - \frac{2 65 + 2}{4} x^{2} = \frac{- 65 + 1}{8}

Since the left-hand side is always positive or 0,and the right-hand side is always negative,the statement is false for any value of x

x = \frac{2 65 + 2}{4} x = - \frac{2 65 + 2}{4} x \in / R

Find the union

x = \frac{2 65 + 2}{4} x = - \frac{2 65 + 2}{4}

Determine the test intervals using the critical values

x < - \frac{2 65 + 2}{4} - \frac{2 65 + 2}{4} < x < \frac{2 65 + 2}{4} x > \frac{2 65 + 2}{4}

Choose a value form each interval

x_{1} = - 2 x_{2} = 0 x_{3} = 2

To determine if x < - \frac{2 65 + 2}{4} is the solution to the inequality,test if the chosen value x = - 2 satisfies the initial inequality

More Steps

x < - \frac{2 65 + 2}{4} is not a solution x_{2} = 0 x_{3} = 2

To determine if - \frac{2 65 + 2}{4} < x < \frac{2 65 + 2}{4} is the solution to the inequality,test if the chosen value x = 0 satisfies the initial inequality

More Steps

x < - \frac{2 65 + 2}{4} is not a solution - \frac{2 65 + 2}{4} < x < \frac{2 65 + 2}{4} is the solution x_{3} = 2

To determine if x > \frac{2 65 + 2}{4} is the solution to the inequality,test if the chosen value x = 2 satisfies the initial inequality

More Steps

x < - \frac{2 65 + 2}{4} is not a solution - \frac{2 65 + 2}{4} < x < \frac{2 65 + 2}{4} is the solution x > \frac{2 65 + 2}{4} is not a solution

The original inequality is a nonstrict inequality,so include the critical value in the solution

- \frac{2 65 + 2}{4} \leq x \leq \frac{2 65 + 2}{4} is the solution

Solution

- \frac{2 65 + 2}{4} \leq x \leq \frac{2 65 + 2}{4}

Alternative Form

x \in - \frac{2 65 + 2}{4}, \frac{2 65 + 2}{4}

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