Solve x^2-2/x-5=3

$$\frac { x ^ { 2 } - 2 } { x - 5 } = 3$$

Solve for x (complex solution)

$$x=\frac{3+\sqrt{43}i}{2}\approx 1.5+3.278719262i$$

$$x=\frac{-\sqrt{43}i+3}{2}\approx 1.5-3.278719262i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 5 since division by zero is not defined. Multiply both sides of the equation by x-5.

$$x^{2}-2=3\left(x-5\right)$$

Use the distributive property to multiply 3 by x-5.

$$x^{2}-2=3x-15$$

Subtract 3x from both sides.

$$x^{2}-2-3x=-15$$

Add 15 to both sides.

$$x^{2}-2-3x+15=0$$

Add -2 and 15 to get 13.

$$x^{2}+13-3x=0$$

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.

$$x^{2}-3x+13=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and 13 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 13}}{2}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 13}}{2}$$

Multiply -4 times 13.

$$x=\frac{-\left(-3\right)±\sqrt{9-52}}{2}$$

Add 9 to -52.

$$x=\frac{-\left(-3\right)±\sqrt{-43}}{2}$$

Take the square root of -43.

$$x=\frac{-\left(-3\right)±\sqrt{43}i}{2}$$

The opposite of -3 is 3.

$$x=\frac{3±\sqrt{43}i}{2}$$

Now solve the equation x=\frac{3±\sqrt{43}i}{2} when ± is plus. Add 3 to i\sqrt{43}.

$$x=\frac{3+\sqrt{43}i}{2}$$

Now solve the equation x=\frac{3±\sqrt{43}i}{2} when ± is minus. Subtract i\sqrt{43} from 3.

$$x=\frac{-\sqrt{43}i+3}{2}$$

The equation is now solved.

$$x=\frac{3+\sqrt{43}i}{2} x=\frac{-\sqrt{43}i+3}{2}$$

Steps for Completing the Square

Variable x cannot be equal to 5 since division by zero is not defined. Multiply both sides of the equation by x-5.

$$x^{2}-2=3\left(x-5\right)$$

Use the distributive property to multiply 3 by x-5.

$$x^{2}-2=3x-15$$

Subtract 3x from both sides.

$$x^{2}-2-3x=-15$$

Add 2 to both sides.

$$x^{2}-3x=-15+2$$

Add -15 and 2 to get -13.

$$x^{2}-3x=-13$$

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

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=-13+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3x+\frac{9}{4}=-13+\frac{9}{4}$$

Add -13 to \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=-\frac{43}{4}$$

Factor x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=-\frac{43}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{-\frac{43}{4}}$$

Simplify.

$$x-\frac{3}{2}=\frac{\sqrt{43}i}{2} x-\frac{3}{2}=-\frac{\sqrt{43}i}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$x=\frac{3+\sqrt{43}i}{2} x=\frac{-\sqrt{43}i+3}{2}$$