Solve x+(1)/(x)=(sqrt(22)-2)/(3) | Equatix - Math Solver

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Example

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

a⋅(b-2)=3b

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

a⋅(b-2)=3b

Question

$$x+ \frac{ 1 }{ x } = \frac{ \sqrt{ 22 } -2 }{ 3 }$$

Solve for x (complex solution)

$x=\frac{i\sqrt{4\sqrt{22}+10}}{6}+\frac{\sqrt{22}}{6}-\frac{1}{3}\approx 0.448402627+0.893831687i$
$x=-\frac{i\sqrt{4\sqrt{22}+10}}{6}+\frac{\sqrt{22}}{6}-\frac{1}{3}\approx 0.448402627-0.893831687i$

Steps Using the Quadratic Formula

Steps for Completing the Square

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 $3x$, the least common multiple of $x,3$.

$$3xx+3=x\left(\sqrt{22}-2\right)$$

Multiply $x$ and $x$ to get $x^{2}$.

$$3x^{2}+3=x\left(\sqrt{22}-2\right)$$

Use the distributive property to multiply $x$ by $\sqrt{22}-2$.

$$3x^{2}+3=x\sqrt{22}-2x$$

Subtract $x\sqrt{22}$ from both sides.

$$3x^{2}+3-x\sqrt{22}=-2x$$

Add $2x$ to both sides.

$$3x^{2}+3-x\sqrt{22}+2x=0$$

Reorder the terms.

$$3x^{2}-\sqrt{22}x+2x+3=0$$

Combine all terms containing $x$.

$$3x^{2}+\left(-\sqrt{22}+2\right)x+3=0$$

All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. The quadratic formula gives two solutions, one when $±$ is addition and one when it is subtraction.

$$3x^{2}+\left(2-\sqrt{22}\right)x+3=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $3$ for $a$, $-\sqrt{22}+2$ for $b$, and $3$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-\left(2-\sqrt{22}\right)±\sqrt{\left(2-\sqrt{22}\right)^{2}-4\times 3\times 3}}{2\times 3}$$

Square $-\sqrt{22}+2$.

$$x=\frac{-\left(2-\sqrt{22}\right)±\sqrt{26-4\sqrt{22}-4\times 3\times 3}}{2\times 3}$$

Multiply $-4$ times $3$.

$$x=\frac{-\left(2-\sqrt{22}\right)±\sqrt{26-4\sqrt{22}-12\times 3}}{2\times 3}$$

Multiply $-12$ times $3$.

$$x=\frac{-\left(2-\sqrt{22}\right)±\sqrt{26-4\sqrt{22}-36}}{2\times 3}$$

Add $26-4\sqrt{22}$ to $-36$.

$$x=\frac{-\left(2-\sqrt{22}\right)±\sqrt{-4\sqrt{22}-10}}{2\times 3}$$

Take the square root of $-10-4\sqrt{22}$.

$$x=\frac{-\left(2-\sqrt{22}\right)±i\sqrt{4\sqrt{22}+10}}{2\times 3}$$

The opposite of $-\sqrt{22}+2$ is $\sqrt{22}-2$.

$$x=\frac{\sqrt{22}-2±i\sqrt{4\sqrt{22}+10}}{2\times 3}$$

Multiply $2$ times $3$.

$$x=\frac{\sqrt{22}-2±i\sqrt{4\sqrt{22}+10}}{6}$$

Now solve the equation $x=\frac{\sqrt{22}-2±i\sqrt{4\sqrt{22}+10}}{6}$ when $±$ is plus. Add $\sqrt{22}-2$ to $i\sqrt{10+4\sqrt{22}}$.

$$x=\frac{\sqrt{22}-2+i\sqrt{4\sqrt{22}+10}}{6}$$

Divide $\sqrt{22}-2+i\sqrt{10+4\sqrt{22}}$ by $6$.

$$x=\frac{i\sqrt{4\sqrt{22}+10}}{6}+\frac{\sqrt{22}}{6}-\frac{1}{3}$$

Now solve the equation $x=\frac{\sqrt{22}-2±i\sqrt{4\sqrt{22}+10}}{6}$ when $±$ is minus. Subtract $i\sqrt{10+4\sqrt{22}}$ from $\sqrt{22}-2$.

$$x=\frac{-i\sqrt{4\sqrt{22}+10}+\sqrt{22}-2}{6}$$

Divide $\sqrt{22}-2-i\sqrt{10+4\sqrt{22}}$ by $6$.

$$x=-\frac{i\sqrt{4\sqrt{22}+10}}{6}+\frac{\sqrt{22}}{6}-\frac{1}{3}$$

The equation is now solved.

$$x=\frac{i\sqrt{4\sqrt{22}+10}}{6}+\frac{\sqrt{22}}{6}-\frac{1}{3}$$ $$x=-\frac{i\sqrt{4\sqrt{22}+10}}{6}+\frac{\sqrt{22}}{6}-\frac{1}{3}$$

Steps for Completing the Square

Variable $x$ cannot be equal to $0$ since division by zero is not defined. Multiply both sides of the equation by $3x$, the least common multiple of $x,3$.

$$3xx+3=x\left(\sqrt{22}-2\right)$$

Multiply $x$ and $x$ to get $x^{2}$.

$$3x^{2}+3=x\left(\sqrt{22}-2\right)$$

Use the distributive property to multiply $x$ by $\sqrt{22}-2$.

$$3x^{2}+3=x\sqrt{22}-2x$$

Subtract $x\sqrt{22}$ from both sides.

$$3x^{2}+3-x\sqrt{22}=-2x$$

Add $2x$ to both sides.

$$3x^{2}+3-x\sqrt{22}+2x=0$$

Subtract $3$ from both sides. Anything subtracted from zero gives its negation.

$$3x^{2}-x\sqrt{22}+2x=-3$$

Combine all terms containing $x$.

$$3x^{2}+\left(-\sqrt{22}+2\right)x=-3$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form $x^{2}+bx=c$.

$$3x^{2}+\left(2-\sqrt{22}\right)x=-3$$

Divide both sides by $3$.

$$\frac{3x^{2}+\left(2-\sqrt{22}\right)x}{3}=-\frac{3}{3}$$

Dividing by $3$ undoes the multiplication by $3$.

$$x^{2}+\frac{2-\sqrt{22}}{3}x=-\frac{3}{3}$$

Divide $-3$ by $3$.

$$x^{2}+\frac{2-\sqrt{22}}{3}x=-1$$

Divide $\frac{-\sqrt{22}+2}{3}$, the coefficient of the $x$ term, by $2$ to get $-\frac{\sqrt{22}}{6}+\frac{1}{3}$. Then add the square of $-\frac{\sqrt{22}}{6}+\frac{1}{3}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{2-\sqrt{22}}{3}x+\left(-\frac{\sqrt{22}}{6}+\frac{1}{3}\right)^{2}=-1+\left(-\frac{\sqrt{22}}{6}+\frac{1}{3}\right)^{2}$$

Square $-\frac{\sqrt{22}}{6}+\frac{1}{3}$.

$$x^{2}+\frac{2-\sqrt{22}}{3}x-\frac{\sqrt{22}}{9}+\frac{13}{18}=-1-\frac{\sqrt{22}}{9}+\frac{13}{18}$$

Add $-1$ to $\frac{13}{18}-\frac{\sqrt{22}}{9}$.

$$x^{2}+\frac{2-\sqrt{22}}{3}x-\frac{\sqrt{22}}{9}+\frac{13}{18}=-\frac{\sqrt{22}}{9}-\frac{5}{18}$$

Factor $x^{2}+\frac{2-\sqrt{22}}{3}x-\frac{\sqrt{22}}{9}+\frac{13}{18}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(x-\frac{\sqrt{22}}{6}+\frac{1}{3}\right)^{2}=-\frac{\sqrt{22}}{9}-\frac{5}{18}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{\sqrt{22}}{6}+\frac{1}{3}\right)^{2}}=\sqrt{-\frac{\sqrt{22}}{9}-\frac{5}{18}}$$

Simplify.

$$x-\frac{\sqrt{22}}{6}+\frac{1}{3}=\frac{i\sqrt{4\sqrt{22}+10}}{6}$$ $$x-\frac{\sqrt{22}}{6}+\frac{1}{3}=-\frac{i\sqrt{4\sqrt{22}+10}}{6}$$

Subtract $-\frac{\sqrt{22}}{6}+\frac{1}{3}$ from both sides of the equation.

$$x=\frac{i\sqrt{4\sqrt{22}+10}}{6}+\frac{\sqrt{22}}{6}-\frac{1}{3}$$ $$x=-\frac{i\sqrt{4\sqrt{22}+10}}{6}+\frac{\sqrt{22}}{6}-\frac{1}{3}$$