Solve x^2-1/x=2269

$$\frac { x ^ { 2 } - 1 } { x } = 2269$$

Solve for x

$$x = \frac{\sqrt{5148365} + 2269}{2} \approx 2269.000440723$$

$$x=\frac{2269-\sqrt{5148365}}{2}\approx -0.000440723$$

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.

$$x^{2}-1=2269x$$

Subtract 2269x from both sides.

$$x^{2}-1-2269x=0$$

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.

$$x^{2}-2269x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2269 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2269\right)±\sqrt{\left(-2269\right)^{2}-4\left(-1\right)}}{2}$$

Square -2269.

$$x=\frac{-\left(-2269\right)±\sqrt{5148361-4\left(-1\right)}}{2}$$

Multiply -4 times -1.

$$x=\frac{-\left(-2269\right)±\sqrt{5148361+4}}{2}$$

Add 5148361 to 4.

$$x=\frac{-\left(-2269\right)±\sqrt{5148365}}{2}$$

The opposite of -2269 is 2269.

$$x=\frac{2269±\sqrt{5148365}}{2}$$

Now solve the equation x=\frac{2269±\sqrt{5148365}}{2} when ± is plus. Add 2269 to \sqrt{5148365}.

$$x=\frac{\sqrt{5148365}+2269}{2}$$

Now solve the equation x=\frac{2269±\sqrt{5148365}}{2} when ± is minus. Subtract \sqrt{5148365} from 2269.

$$x=\frac{2269-\sqrt{5148365}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{5148365}+2269}{2} x=\frac{2269-\sqrt{5148365}}{2}$$

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.

$$x^{2}-1=2269x$$

Subtract 2269x from both sides.

$$x^{2}-1-2269x=0$$

Add 1 to both sides. Anything plus zero gives itself.

$$x^{2}-2269x=1$$

Divide -2269, the coefficient of the x term, by 2 to get -\frac{2269}{2}. Then add the square of -\frac{2269}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2269x+\left(-\frac{2269}{2}\right)^{2}=1+\left(-\frac{2269}{2}\right)^{2}$$

Square -\frac{2269}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-2269x+\frac{5148361}{4}=1+\frac{5148361}{4}$$

Add 1 to \frac{5148361}{4}.

$$x^{2}-2269x+\frac{5148361}{4}=\frac{5148365}{4}$$

Factor x^{2}-2269x+\frac{5148361}{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}.

$$\left(x-\frac{2269}{2}\right)^{2}=\frac{5148365}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{2269}{2}\right)^{2}}=\sqrt{\frac{5148365}{4}}$$

Simplify.

$$x-\frac{2269}{2}=\frac{\sqrt{5148365}}{2} x-\frac{2269}{2}=-\frac{\sqrt{5148365}}{2}$$

Add \frac{2269}{2} to both sides of the equation.

$$x=\frac{\sqrt{5148365}+2269}{2} x=\frac{2269-\sqrt{5148365}}{2}$$