Solve x^2/x-1=1000

$$\frac { x ^ { 2 } } { x - 1 } = 1000$$

Solve for x

$$x=10\sqrt{2490}+500\approx 998.998997995$$

$$x=500-10\sqrt{2490}\approx 1.001002005$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by x-1.

$$x^{2}=1000\left(x-1\right)$$

Use the distributive property to multiply 1000 by x-1.

$$x^{2}=1000x-1000$$

Subtract 1000x from both sides.

$$x^{2}-1000x=-1000$$

Add 1000 to both sides.

$$x^{2}-1000x+1000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1000 for b, and 1000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1000\right)±\sqrt{\left(-1000\right)^{2}-4\times 1000}}{2}$$

Square -1000.

$$x=\frac{-\left(-1000\right)±\sqrt{1000000-4\times 1000}}{2}$$

Multiply -4 times 1000.

$$x=\frac{-\left(-1000\right)±\sqrt{1000000-4000}}{2}$$

Add 1000000 to -4000.

$$x=\frac{-\left(-1000\right)±\sqrt{996000}}{2}$$

Take the square root of 996000.

$$x=\frac{-\left(-1000\right)±20\sqrt{2490}}{2}$$

The opposite of -1000 is 1000.

$$x=\frac{1000±20\sqrt{2490}}{2}$$

Now solve the equation x=\frac{1000±20\sqrt{2490}}{2} when ± is plus. Add 1000 to 20\sqrt{2490}.

$$x=\frac{20\sqrt{2490}+1000}{2}$$

Divide 1000+20\sqrt{2490} by 2.

$$x=10\sqrt{2490}+500$$

Now solve the equation x=\frac{1000±20\sqrt{2490}}{2} when ± is minus. Subtract 20\sqrt{2490} from 1000.

$$x=\frac{1000-20\sqrt{2490}}{2}$$

Divide 1000-20\sqrt{2490} by 2.

$$x=500-10\sqrt{2490}$$

The equation is now solved.

$$x=10\sqrt{2490}+500 x=500-10\sqrt{2490}$$

Steps for Completing the Square

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by x-1.

$$x^{2}=1000\left(x-1\right)$$

Use the distributive property to multiply 1000 by x-1.

$$x^{2}=1000x-1000$$

Subtract 1000x from both sides.

$$x^{2}-1000x=-1000$$

Divide -1000, the coefficient of the x term, by 2 to get -500. Then add the square of -500 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-1000x+\left(-500\right)^{2}=-1000+\left(-500\right)^{2}$$

Square -500.

$$x^{2}-1000x+250000=-1000+250000$$

Add -1000 to 250000.

$$x^{2}-1000x+250000=249000$$

Factor x^{2}-1000x+250000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-500\right)^{2}=249000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-500\right)^{2}}=\sqrt{249000}$$

Simplify.

$$x-500=10\sqrt{2490} x-500=-10\sqrt{2490}$$

Add 500 to both sides of the equation.

$$x=10\sqrt{2490}+500 x=500-10\sqrt{2490}$$