Solve x+(100/x-1)=14

$$x+( \frac{ 100 }{ x } -1) = 14$$

Solve for x (complex solution)

$$x=\frac{15+5\sqrt{7}i}{2}\approx 7.5+6.614378278i$$

$$x=\frac{-5\sqrt{7}i+15}{2}\approx 7.5-6.614378278i$$

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.

$$xx+100+x\left(-1\right)=14x$$

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

$$x^{2}+100+x\left(-1\right)=14x$$

Subtract 14x from both sides.

$$x^{2}+100+x\left(-1\right)-14x=0$$

Combine x\left(-1\right) and -14x to get -15x.

$$x^{2}+100-15x=0$$

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.

$$x^{2}-15x+100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -15 for b, and 100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 100}}{2}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 100}}{2}$$

Multiply -4 times 100.

$$x=\frac{-\left(-15\right)±\sqrt{225-400}}{2}$$

Add 225 to -400.

$$x=\frac{-\left(-15\right)±\sqrt{-175}}{2}$$

Take the square root of -175.

$$x=\frac{-\left(-15\right)±5\sqrt{7}i}{2}$$

The opposite of -15 is 15.

$$x=\frac{15±5\sqrt{7}i}{2}$$

Now solve the equation x=\frac{15±5\sqrt{7}i}{2} when ± is plus. Add 15 to 5i\sqrt{7}.

$$x=\frac{15+5\sqrt{7}i}{2}$$

Now solve the equation x=\frac{15±5\sqrt{7}i}{2} when ± is minus. Subtract 5i\sqrt{7} from 15.

$$x=\frac{-5\sqrt{7}i+15}{2}$$

The equation is now solved.

$$x=\frac{15+5\sqrt{7}i}{2} x=\frac{-5\sqrt{7}i+15}{2}$$

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.

$$xx+100+x\left(-1\right)=14x$$

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

$$x^{2}+100+x\left(-1\right)=14x$$

Subtract 14x from both sides.

$$x^{2}+100+x\left(-1\right)-14x=0$$

Combine x\left(-1\right) and -14x to get -15x.

$$x^{2}+100-15x=0$$

Subtract 100 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}-15x=-100$$

Divide -15, the coefficient of the x term, by 2 to get -\frac{15}{2}. Then add the square of -\frac{15}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=-100+\left(-\frac{15}{2}\right)^{2}$$

Square -\frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-15x+\frac{225}{4}=-100+\frac{225}{4}$$

Add -100 to \frac{225}{4}.

$$x^{2}-15x+\frac{225}{4}=-\frac{175}{4}$$

Factor x^{2}-15x+\frac{225}{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}.

$$\left(x-\frac{15}{2}\right)^{2}=-\frac{175}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{-\frac{175}{4}}$$

Simplify.

$$x-\frac{15}{2}=\frac{5\sqrt{7}i}{2} x-\frac{15}{2}=-\frac{5\sqrt{7}i}{2}$$

Add \frac{15}{2} to both sides of the equation.

$$x=\frac{15+5\sqrt{7}i}{2} x=\frac{-5\sqrt{7}i+15}{2}$$