How to solve 2 - 5x = 4/x

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 - 5x = 4/x | Equatix

Question

$$2-5x=\frac{4}{x}$$

Solve for x (complex solution)

$x=\frac{-\sqrt{19}i+1}{5}\approx 0.2-0.871779789i$
$x=\frac{1+\sqrt{19}i}{5}\approx 0.2+0.871779789i$

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 $x$.

$$x\times 2-5xx=4$$

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

$$x\times 2-5x^{2}=4$$

Subtract $4$ from both sides.

$$x\times 2-5x^{2}-4=0$$

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.

$$-5x^{2}+2x-4=0$$

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

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

Square $2$.

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

Multiply $-4$ times $-5$.

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

Multiply $20$ times $-4$.

$$x=\frac{-2±\sqrt{4-80}}{2\left(-5\right)}$$

Add $4$ to $-80$.

$$x=\frac{-2±\sqrt{-76}}{2\left(-5\right)}$$

Take the square root of $-76$.

$$x=\frac{-2±2\sqrt{19}i}{2\left(-5\right)}$$

Multiply $2$ times $-5$.

$$x=\frac{-2±2\sqrt{19}i}{-10}$$

Now solve the equation $x=\frac{-2±2\sqrt{19}i}{-10}$ when $±$ is plus. Add $-2$ to $2i\sqrt{19}$.

$$x=\frac{-2+2\sqrt{19}i}{-10}$$

Divide $-2+2i\sqrt{19}$ by $-10$.

$$x=\frac{-\sqrt{19}i+1}{5}$$

Now solve the equation $x=\frac{-2±2\sqrt{19}i}{-10}$ when $±$ is minus. Subtract $2i\sqrt{19}$ from $-2$.

$$x=\frac{-2\sqrt{19}i-2}{-10}$$

Divide $-2-2i\sqrt{19}$ by $-10$.

$$x=\frac{1+\sqrt{19}i}{5}$$

The equation is now solved.

$$x=\frac{-\sqrt{19}i+1}{5}$$ $$x=\frac{1+\sqrt{19}i}{5}$$

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 $x$.

$$x\times 2-5xx=4$$

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

$$x\times 2-5x^{2}=4$$

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

$$-5x^{2}+2x=4$$

Divide both sides by $-5$.

$$\frac{-5x^{2}+2x}{-5}=\frac{4}{-5}$$

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

$$x^{2}+\frac{2}{-5}x=\frac{4}{-5}$$

Divide $2$ by $-5$.

$$x^{2}-\frac{2}{5}x=\frac{4}{-5}$$

Divide $4$ by $-5$.

$$x^{2}-\frac{2}{5}x=-\frac{4}{5}$$

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.

$$x^{2}-\frac{2}{5}x+\left(-\frac{1}{5}\right)^{2}=-\frac{4}{5}+\left(-\frac{1}{5}\right)^{2}$$

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

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=-\frac{4}{5}+\frac{1}{25}$$

Add $-\frac{4}{5}$ to $\frac{1}{25}$ by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=-\frac{19}{25}$$

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 $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(x-\frac{1}{5}\right)^{2}=-\frac{19}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{5}\right)^{2}}=\sqrt{-\frac{19}{25}}$$

Simplify.

$$x-\frac{1}{5}=\frac{\sqrt{19}i}{5}$$ $$x-\frac{1}{5}=-\frac{\sqrt{19}i}{5}$$

Add $\frac{1}{5}$ to both sides of the equation.

$$x=\frac{1+\sqrt{19}i}{5}$$ $$x=\frac{-\sqrt{19}i+1}{5}$$

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Example

Question

Solve for x (complex solution)

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using the Quadratic Formula

Steps for Completing the Square