Solve x/4*2=2000-x/x-1

$$\frac { x } { 4 } \times 2 = \frac { 2000 - x } { x - 1 }$$

Solve for x

$$x = \frac{\sqrt{16001} - 1}{2} \approx 62.747529596$$

$$x=\frac{-\sqrt{16001}-1}{2}\approx -63.747529596$$

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 4\left(x-1\right), the least common multiple of 4,x-1.

$$\left(x-1\right)x\times 2=4\left(2000-x\right)$$

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

$$\left(x^{2}-x\right)\times 2=4\left(2000-x\right)$$

Use the distributive property to multiply x^{2}-x by 2.

$$2x^{2}-2x=4\left(2000-x\right)$$

Use the distributive property to multiply 4 by 2000-x.

$$2x^{2}-2x=8000-4x$$

Subtract 8000 from both sides.

$$2x^{2}-2x-8000=-4x$$

Add 4x to both sides.

$$2x^{2}-2x-8000+4x=0$$

Combine -2x and 4x to get 2x.

$$2x^{2}+2x-8000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 2 for b, and -8000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\times 2\left(-8000\right)}}{2\times 2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\times 2\left(-8000\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-2±\sqrt{4-8\left(-8000\right)}}{2\times 2}$$

Multiply -8 times -8000.

$$x=\frac{-2±\sqrt{4+64000}}{2\times 2}$$

Add 4 to 64000.

$$x=\frac{-2±\sqrt{64004}}{2\times 2}$$

Take the square root of 64004.

$$x=\frac{-2±2\sqrt{16001}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-2±2\sqrt{16001}}{4}$$

Now solve the equation x=\frac{-2±2\sqrt{16001}}{4} when ± is plus. Add -2 to 2\sqrt{16001}.

$$x=\frac{2\sqrt{16001}-2}{4}$$

Divide -2+2\sqrt{16001} by 4.

$$x=\frac{\sqrt{16001}-1}{2}$$

Now solve the equation x=\frac{-2±2\sqrt{16001}}{4} when ± is minus. Subtract 2\sqrt{16001} from -2.

$$x=\frac{-2\sqrt{16001}-2}{4}$$

Divide -2-2\sqrt{16001} by 4.

$$x=\frac{-\sqrt{16001}-1}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{16001}-1}{2} x=\frac{-\sqrt{16001}-1}{2}$$

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 4\left(x-1\right), the least common multiple of 4,x-1.

$$\left(x-1\right)x\times 2=4\left(2000-x\right)$$

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

$$\left(x^{2}-x\right)\times 2=4\left(2000-x\right)$$

Use the distributive property to multiply x^{2}-x by 2.

$$2x^{2}-2x=4\left(2000-x\right)$$

Use the distributive property to multiply 4 by 2000-x.

$$2x^{2}-2x=8000-4x$$

Add 4x to both sides.

$$2x^{2}-2x+4x=8000$$

Combine -2x and 4x to get 2x.

$$2x^{2}+2x=8000$$

Divide both sides by 2.

$$\frac{2x^{2}+2x}{2}=\frac{8000}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{2}{2}x=\frac{8000}{2}$$

Divide 2 by 2.

$$x^{2}+x=\frac{8000}{2}$$

Divide 8000 by 2.

$$x^{2}+x=4000$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=4000+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=4000+\frac{1}{4}$$

Add 4000 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{16001}{4}$$

Factor x^{2}+x+\frac{1}{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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{16001}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{16001}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{16001}}{2} x+\frac{1}{2}=-\frac{\sqrt{16001}}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{\sqrt{16001}-1}{2} x=\frac{-\sqrt{16001}-1}{2}$$