How to solve 2/x = 5x + 2

1. Factor the equation (x + 2)(x + 3) = 0. 2. Set each factor equal to zero: x + 2 = 0 and x + 3 = 0. 3. Solve for x: x = -2 and x = -3.

Solve 2/x = 5x + 2 | Equatix

Question

$$\frac{2}{x}=5x+2$$

Solve for x

$x=\frac{\sqrt{11}-1}{5}\approx 0.463324958$
$x=\frac{-\sqrt{11}-1}{5}\approx -0.863324958$

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 $x$.

$$2=5xx+x\times 2$$

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

$$2=5x^{2}+x\times 2$$

Swap sides so that all variable terms are on the left hand side.

$$5x^{2}+x\times 2=2$$

Subtract $2$ from both sides.

$$5x^{2}+x\times 2-2=0$$

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.

$$5x^{2}+2x-2=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $5$ for $a$, $2$ for $b$, and $-2$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-2±\sqrt{2^{2}-4\times 5\left(-2\right)}}{2\times 5}$$

Square $2$.

$$x=\frac{-2±\sqrt{4-4\times 5\left(-2\right)}}{2\times 5}$$

Multiply $-4$ times $5$.

$$x=\frac{-2±\sqrt{4-20\left(-2\right)}}{2\times 5}$$

Multiply $-20$ times $-2$.

$$x=\frac{-2±\sqrt{4+40}}{2\times 5}$$

Add $4$ to $40$.

$$x=\frac{-2±\sqrt{44}}{2\times 5}$$

Take the square root of $44$.

$$x=\frac{-2±2\sqrt{11}}{2\times 5}$$

Multiply $2$ times $5$.

$$x=\frac{-2±2\sqrt{11}}{10}$$

Now solve the equation $x=\frac{-2±2\sqrt{11}}{10}$ when $±$ is plus. Add $-2$ to $2\sqrt{11}$.

$$x=\frac{2\sqrt{11}-2}{10}$$

Divide $-2+2\sqrt{11}$ by $10$.

$$x=\frac{\sqrt{11}-1}{5}$$

Now solve the equation $x=\frac{-2±2\sqrt{11}}{10}$ when $±$ is minus. Subtract $2\sqrt{11}$ from $-2$.

$$x=\frac{-2\sqrt{11}-2}{10}$$

Divide $-2-2\sqrt{11}$ by $10$.

$$x=\frac{-\sqrt{11}-1}{5}$$

The equation is now solved.

$$x=\frac{\sqrt{11}-1}{5}$$ $$x=\frac{-\sqrt{11}-1}{5}$$

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 $x$.

$$2=5xx+x\times 2$$

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

$$2=5x^{2}+x\times 2$$

Swap sides so that all variable terms are on the left hand side.

$$5x^{2}+x\times 2=2$$

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 $x^{2}+bx=c$.

$$5x^{2}+2x=2$$

Divide both sides by $5$.

$$\frac{5x^{2}+2x}{5}=\frac{2}{5}$$

Dividing by $5$ undoes the multiplication by $5$.

$$x^{2}+\frac{2}{5}x=\frac{2}{5}$$

Divide $\frac{2}{5}$, the coefficient of the $x$ term, by $2$ to get $\frac{1}{5}$. Then add the square of $\frac{1}{5}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{2}{5}x+\left(\frac{1}{5}\right)^{2}=\frac{2}{5}+\left(\frac{1}{5}\right)^{2}$$

Square $\frac{1}{5}$ by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{2}{5}x+\frac{1}{25}=\frac{2}{5}+\frac{1}{25}$$

Add $\frac{2}{5}$ to $\frac{1}{25}$ by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{2}{5}x+\frac{1}{25}=\frac{11}{25}$$

Factor $x^{2}+\frac{2}{5}x+\frac{1}{25}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(x+\frac{1}{5}\right)^{2}=\frac{11}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{5}\right)^{2}}=\sqrt{\frac{11}{25}}$$

Simplify.

$$x+\frac{1}{5}=\frac{\sqrt{11}}{5}$$ $$x+\frac{1}{5}=-\frac{\sqrt{11}}{5}$$

Subtract $\frac{1}{5}$ from both sides of the equation.

$$x=\frac{\sqrt{11}-1}{5}$$ $$x=\frac{-\sqrt{11}-1}{5}$$

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Solve for x

Steps Using the Quadratic Formula

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Steps Using the Quadratic Formula

Steps for Completing the Square