Solve (x)^(2)+(1)/((x)^(2))=18 | Equatix - Math Solver

Fractions & Decimals

Ratios & Proportions

Basic Number Theory

Integers & Absolute Values

Factors & Multiples

Prime Factorization

Basic Equations & Inequalities

Coordinate Plane & Graphing

Linear Equations & Inequalities

Quadratic Equations

Polynomials & Factoring

Functions & Graphs

Systems of Equations

Trigonometric Functions

Trigonometric Identities & Equations

Right Triangle Trigonometry

Graphing Trigonometric Functions

Limits & Continuity

Derivatives & Applications

Integrals & Applications

Series & Sequences

Multivariable Calculus

Functions & Graphs

Polynomial & Rational Functions

Exponential & Logarithmic Functions

Sequences & Series

Descriptive Statistics

Normal Distribution

Hypothesis Testing

Regression Analysis

Linear Programming

Sets & Counting

Eigenvalues & Eigenvectors

Linear Transformations

Atomic Structure

Thermochemistry

Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$${ x }^{ 2 } + \frac{ 1 }{ { x }^{ 2 } } =18$$

Solve for x

$x=-\left(\sqrt{5}+2\right)\approx -4.236067977$
$x=\sqrt{5}+2\approx 4.236067977$
$x=\sqrt{5}-2\approx 0.236067977$
$x=2-\sqrt{5}\approx -0.236067977$

Steps Using the Quadratic Formula

Variable $x$ cannot be equal to $0$ since division by zero is not defined. Multiply both sides of the equation by $x^{2}$.

$$x^{2}x^{2}+1=18x^{2}$$

To multiply powers of the same base, add their exponents. Add $2$ and $2$ to get $4$.

$$x^{4}+1=18x^{2}$$

Subtract $18x^{2}$ from both sides.

$$x^{4}+1-18x^{2}=0$$

Substitute $t$ for $x^{2}$.

$$t^{2}-18t+1=0$$

All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. Substitute $1$ for $a$, $-18$ for $b$, and $1$ for $c$ in the quadratic formula.

$$t=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 1\times 1}}{2}$$

Do the calculations.

$$t=\frac{18±8\sqrt{5}}{2}$$

Solve the equation $t=\frac{18±8\sqrt{5}}{2}$ when $±$ is plus and when $±$ is minus.

$$t=4\sqrt{5}+9$$ $$t=9-4\sqrt{5}$$

Since $x=t^{2}$, the solutions are obtained by evaluating $x=±\sqrt{t}$ for each $t$.

$$x=\sqrt{5}+2$$ $$x=-\left(\sqrt{5}+2\right)$$ $$x=-\left(2-\sqrt{5}\right)$$ $$x=2-\sqrt{5}$$