Solve x=x-1300/100-x-3

$$x= \frac{ x-1300 }{ 100-x } -3$$

Solve for x

$$x=8\sqrt{61}+48\approx 110.481997407$$

$$x=48-8\sqrt{61}\approx -14.481997407$$

Steps Using the Quadratic Formula

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{100-x}{100-x}.

$$x=\frac{x-1300}{100-x}-\frac{3\left(100-x\right)}{100-x}$$

Since \frac{x-1300}{100-x} and \frac{3\left(100-x\right)}{100-x} have the same denominator, subtract them by subtracting their numerators.

$$x=\frac{x-1300-3\left(100-x\right)}{100-x}$$

Do the multiplications in x-1300-3\left(100-x\right).

$$x=\frac{x-1300-300+3x}{100-x}$$

Combine like terms in x-1300-300+3x.

$$x=\frac{4x-1600}{100-x}$$

Subtract \frac{4x-1600}{100-x} from both sides.

$$x-\frac{4x-1600}{100-x}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{100-x}{100-x}.

$$\frac{x\left(100-x\right)}{100-x}-\frac{4x-1600}{100-x}=0$$

Since \frac{x\left(100-x\right)}{100-x} and \frac{4x-1600}{100-x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\left(100-x\right)-\left(4x-1600\right)}{100-x}=0$$

Do the multiplications in x\left(100-x\right)-\left(4x-1600\right).

$$\frac{100x-x^{2}-4x+1600}{100-x}=0$$

Combine like terms in 100x-x^{2}-4x+1600.

$$\frac{96x-x^{2}+1600}{100-x}=0$$

Variable x cannot be equal to 100 since division by zero is not defined. Multiply both sides of the equation by -x+100.

$$96x-x^{2}+1600=0$$

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.

$$-x^{2}+96x+1600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 96 for b, and 1600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-96±\sqrt{96^{2}-4\left(-1\right)\times 1600}}{2\left(-1\right)}$$

Square 96.

$$x=\frac{-96±\sqrt{9216-4\left(-1\right)\times 1600}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-96±\sqrt{9216+4\times 1600}}{2\left(-1\right)}$$

Multiply 4 times 1600.

$$x=\frac{-96±\sqrt{9216+6400}}{2\left(-1\right)}$$

Add 9216 to 6400.

$$x=\frac{-96±\sqrt{15616}}{2\left(-1\right)}$$

Take the square root of 15616.

$$x=\frac{-96±16\sqrt{61}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-96±16\sqrt{61}}{-2}$$

Now solve the equation x=\frac{-96±16\sqrt{61}}{-2} when ± is plus. Add -96 to 16\sqrt{61}.

$$x=\frac{16\sqrt{61}-96}{-2}$$

Divide -96+16\sqrt{61} by -2.

$$x=48-8\sqrt{61}$$

Now solve the equation x=\frac{-96±16\sqrt{61}}{-2} when ± is minus. Subtract 16\sqrt{61} from -96.

$$x=\frac{-16\sqrt{61}-96}{-2}$$

Divide -96-16\sqrt{61} by -2.

$$x=8\sqrt{61}+48$$

The equation is now solved.

$$x=48-8\sqrt{61} x=8\sqrt{61}+48$$

Steps for Completing the Square

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{100-x}{100-x}.

$$x=\frac{x-1300}{100-x}-\frac{3\left(100-x\right)}{100-x}$$

Since \frac{x-1300}{100-x} and \frac{3\left(100-x\right)}{100-x} have the same denominator, subtract them by subtracting their numerators.

$$x=\frac{x-1300-3\left(100-x\right)}{100-x}$$

Do the multiplications in x-1300-3\left(100-x\right).

$$x=\frac{x-1300-300+3x}{100-x}$$

Combine like terms in x-1300-300+3x.

$$x=\frac{4x-1600}{100-x}$$

Subtract \frac{4x-1600}{100-x} from both sides.

$$x-\frac{4x-1600}{100-x}=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{100-x}{100-x}.

$$\frac{x\left(100-x\right)}{100-x}-\frac{4x-1600}{100-x}=0$$

Since \frac{x\left(100-x\right)}{100-x} and \frac{4x-1600}{100-x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\left(100-x\right)-\left(4x-1600\right)}{100-x}=0$$

Do the multiplications in x\left(100-x\right)-\left(4x-1600\right).

$$\frac{100x-x^{2}-4x+1600}{100-x}=0$$

Combine like terms in 100x-x^{2}-4x+1600.

$$\frac{96x-x^{2}+1600}{100-x}=0$$

Variable x cannot be equal to 100 since division by zero is not defined. Multiply both sides of the equation by -x+100.

$$96x-x^{2}+1600=0$$

Subtract 1600 from both sides. Anything subtracted from zero gives its negation.

$$96x-x^{2}=-1600$$

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.

$$-x^{2}+96x=-1600$$

Divide both sides by -1.

$$\frac{-x^{2}+96x}{-1}=-\frac{1600}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{96}{-1}x=-\frac{1600}{-1}$$

Divide 96 by -1.

$$x^{2}-96x=-\frac{1600}{-1}$$

Divide -1600 by -1.

$$x^{2}-96x=1600$$

Divide -96, the coefficient of the x term, by 2 to get -48. Then add the square of -48 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-96x+\left(-48\right)^{2}=1600+\left(-48\right)^{2}$$

Square -48.

$$x^{2}-96x+2304=1600+2304$$

Add 1600 to 2304.

$$x^{2}-96x+2304=3904$$

Factor x^{2}-96x+2304. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-48\right)^{2}=3904$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-48\right)^{2}}=\sqrt{3904}$$

Simplify.

$$x-48=8\sqrt{61} x-48=-8\sqrt{61}$$

Add 48 to both sides of the equation.

$$x=8\sqrt{61}+48 x=48-8\sqrt{61}$$