Solve 200divx-1=200-5x/x+5

$$200 \div x-1= \frac{ 200-5x }{ x+5 }$$

Solve for x (complex solution)

$$x=\frac{5+15\sqrt{71}i}{8}\approx 0.625+15.799030825i$$

$$x=\frac{-15\sqrt{71}i+5}{8}\approx 0.625-15.799030825i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -5,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+5\right), the least common multiple of x,x+5.

$$\left(x+5\right)\times 200+x\left(x+5\right)\left(-1\right)=x\left(200-5x\right)$$

Use the distributive property to multiply x+5 by 200.

$$200x+1000+x\left(x+5\right)\left(-1\right)=x\left(200-5x\right)$$

Use the distributive property to multiply x by x+5.

$$200x+1000+\left(x^{2}+5x\right)\left(-1\right)=x\left(200-5x\right)$$

Use the distributive property to multiply x^{2}+5x by -1.

$$200x+1000-x^{2}-5x=x\left(200-5x\right)$$

Combine 200x and -5x to get 195x.

$$195x+1000-x^{2}=x\left(200-5x\right)$$

Use the distributive property to multiply x by 200-5x.

$$195x+1000-x^{2}=200x-5x^{2}$$

Subtract 200x from both sides.

$$195x+1000-x^{2}-200x=-5x^{2}$$

Combine 195x and -200x to get -5x.

$$-5x+1000-x^{2}=-5x^{2}$$

Add 5x^{2} to both sides.

$$-5x+1000-x^{2}+5x^{2}=0$$

Combine -x^{2} and 5x^{2} to get 4x^{2}.

$$-5x+1000+4x^{2}=0$$

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.

$$4x^{2}-5x+1000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -5 for b, and 1000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 4\times 1000}}{2\times 4}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 4\times 1000}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-5\right)±\sqrt{25-16\times 1000}}{2\times 4}$$

Multiply -16 times 1000.

$$x=\frac{-\left(-5\right)±\sqrt{25-16000}}{2\times 4}$$

Add 25 to -16000.

$$x=\frac{-\left(-5\right)±\sqrt{-15975}}{2\times 4}$$

Take the square root of -15975.

$$x=\frac{-\left(-5\right)±15\sqrt{71}i}{2\times 4}$$

The opposite of -5 is 5.

$$x=\frac{5±15\sqrt{71}i}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{5±15\sqrt{71}i}{8}$$

Now solve the equation x=\frac{5±15\sqrt{71}i}{8} when ± is plus. Add 5 to 15i\sqrt{71}.

$$x=\frac{5+15\sqrt{71}i}{8}$$

Now solve the equation x=\frac{5±15\sqrt{71}i}{8} when ± is minus. Subtract 15i\sqrt{71} from 5.

$$x=\frac{-15\sqrt{71}i+5}{8}$$

The equation is now solved.

$$x=\frac{5+15\sqrt{71}i}{8} x=\frac{-15\sqrt{71}i+5}{8}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -5,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+5\right), the least common multiple of x,x+5.

$$\left(x+5\right)\times 200+x\left(x+5\right)\left(-1\right)=x\left(200-5x\right)$$

Use the distributive property to multiply x+5 by 200.

$$200x+1000+x\left(x+5\right)\left(-1\right)=x\left(200-5x\right)$$

Use the distributive property to multiply x by x+5.

$$200x+1000+\left(x^{2}+5x\right)\left(-1\right)=x\left(200-5x\right)$$

Use the distributive property to multiply x^{2}+5x by -1.

$$200x+1000-x^{2}-5x=x\left(200-5x\right)$$

Combine 200x and -5x to get 195x.

$$195x+1000-x^{2}=x\left(200-5x\right)$$

Use the distributive property to multiply x by 200-5x.

$$195x+1000-x^{2}=200x-5x^{2}$$

Subtract 200x from both sides.

$$195x+1000-x^{2}-200x=-5x^{2}$$

Combine 195x and -200x to get -5x.

$$-5x+1000-x^{2}=-5x^{2}$$

Add 5x^{2} to both sides.

$$-5x+1000-x^{2}+5x^{2}=0$$

Combine -x^{2} and 5x^{2} to get 4x^{2}.

$$-5x+1000+4x^{2}=0$$

Subtract 1000 from both sides. Anything subtracted from zero gives its negation.

$$-5x+4x^{2}=-1000$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$4x^{2}-5x=-1000$$

Divide both sides by 4.

$$\frac{4x^{2}-5x}{4}=-\frac{1000}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}-\frac{5}{4}x=-\frac{1000}{4}$$

Divide -1000 by 4.

$$x^{2}-\frac{5}{4}x=-250$$

Divide -\frac{5}{4}, the coefficient of the x term, by 2 to get -\frac{5}{8}. Then add the square of -\frac{5}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{4}x+\left(-\frac{5}{8}\right)^{2}=-250+\left(-\frac{5}{8}\right)^{2}$$

Square -\frac{5}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=-250+\frac{25}{64}$$

Add -250 to \frac{25}{64}.

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=-\frac{15975}{64}$$

Factor x^{2}-\frac{5}{4}x+\frac{25}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{8}\right)^{2}=-\frac{15975}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{8}\right)^{2}}=\sqrt{-\frac{15975}{64}}$$

Simplify.

$$x-\frac{5}{8}=\frac{15\sqrt{71}i}{8} x-\frac{5}{8}=-\frac{15\sqrt{71}i}{8}$$

Add \frac{5}{8} to both sides of the equation.

$$x=\frac{5+15\sqrt{71}i}{8} x=\frac{-15\sqrt{71}i+5}{8}$$