Solve x+4/x-3=e

$$x + \frac { 4 } { x } - 3 = e$$

Solve for x (complex solution)

$$x=\frac{\sqrt{\left(e-1\right)\left(e+7\right)}+e+3}{2}\approx 4.902345938$$

$$x=\frac{-\sqrt{\left(e-1\right)\left(e+7\right)}+e+3}{2}\approx 0.815935891$$

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.

$$xx+4+x\left(-3\right)=ex$$

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

$$x^{2}+4+x\left(-3\right)=ex$$

Subtract ex from both sides.

$$x^{2}+4+x\left(-3\right)-ex=0$$

Reorder the terms.

$$x^{2}-ex-3x+4=0$$

Combine all terms containing x.

$$x^{2}+\left(-e-3\right)x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -e-3 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -e-3.

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

Multiply -4 times 4.

$$x=\frac{-\left(-e-3\right)±\sqrt{\left(e+3\right)^{2}-16}}{2}$$

Add \left(e+3\right)^{2} to -16.

$$x=\frac{-\left(-e-3\right)±\sqrt{\left(e-1\right)\left(e+7\right)}}{2}$$

The opposite of -e-3 is e+3.

$$x=\frac{e+3±\sqrt{\left(e-1\right)\left(e+7\right)}}{2}$$

Now solve the equation x=\frac{e+3±\sqrt{\left(e-1\right)\left(e+7\right)}}{2} when ± is plus. Add e+3 to \sqrt{\left(e+7\right)\left(e-1\right)}.

$$x=\frac{\sqrt{\left(e-1\right)\left(e+7\right)}+e+3}{2}$$

Now solve the equation x=\frac{e+3±\sqrt{\left(e-1\right)\left(e+7\right)}}{2} when ± is minus. Subtract \sqrt{\left(e+7\right)\left(e-1\right)} from e+3.

$$x=\frac{-\sqrt{\left(e-1\right)\left(e+7\right)}+e+3}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{\left(e-1\right)\left(e+7\right)}+e+3}{2} x=\frac{-\sqrt{\left(e-1\right)\left(e+7\right)}+e+3}{2}$$

Variable x cannot be equal to 0.

$$x=\frac{-\sqrt{\left(e-1\right)\left(e+7\right)}+e+3}{2}\text{, }x\neq 0 x=\frac{\sqrt{\left(e-1\right)\left(e+7\right)}+e+3}{2}\text{, }x\neq 0$$

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.

$$xx+4+x\left(-3\right)=ex$$

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

$$x^{2}+4+x\left(-3\right)=ex$$

Subtract ex from both sides.

$$x^{2}+4+x\left(-3\right)-ex=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}+x\left(-3\right)-ex=-4$$

Combine all terms containing x.

$$x^{2}+\left(-3-e\right)x=-4$$

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.

$$x^{2}+\left(-e-3\right)x=-4$$

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

$$x^{2}+\left(-e-3\right)x+\left(\frac{-e-3}{2}\right)^{2}=-4+\left(\frac{-e-3}{2}\right)^{2}$$

Square \frac{-e-3}{2}.

$$x^{2}+\left(-e-3\right)x+\frac{\left(e+3\right)^{2}}{4}=-4+\frac{\left(e+3\right)^{2}}{4}$$

Add -4 to \frac{\left(e+3\right)^{2}}{4}.

$$x^{2}+\left(-e-3\right)x+\frac{\left(e+3\right)^{2}}{4}=\frac{\left(e-1\right)\left(e+7\right)}{4}$$

Factor x^{2}+\left(-e-3\right)x+\frac{\left(e+3\right)^{2}}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{-e-3}{2}\right)^{2}=\frac{\left(e-1\right)\left(e+7\right)}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{-e-3}{2}\right)^{2}}=\sqrt{\frac{\left(e-1\right)\left(e+7\right)}{4}}$$

Simplify.

$$x+\frac{-e-3}{2}=\frac{\sqrt{\left(e-1\right)\left(e+7\right)}}{2} x+\frac{-e-3}{2}=-\frac{\sqrt{\left(e-1\right)\left(e+7\right)}}{2}$$

Subtract \frac{-e-3}{2} from both sides of the equation.

$$x=\frac{\sqrt{\left(e-1\right)\left(e+7\right)}+e+3}{2} x=\frac{-\sqrt{\left(e-1\right)\left(e+7\right)}+e+3}{2}$$

Variable x cannot be equal to 0.

$$x=\frac{-\sqrt{\left(e-1\right)\left(e+7\right)}+e+3}{2}\text{, }x\neq 0 x=\frac{\sqrt{\left(e-1\right)\left(e+7\right)}+e+3}{2}\text{, }x\neq 0$$

Solve for x

$$x=\frac{\sqrt{e^{2}+6e-7}+e+3}{2}\approx 4.902345938$$

$$x=\frac{-\sqrt{e^{2}+6e-7}+e+3}{2}\approx 0.815935891$$