Solve 48800/x=24000x/210-x

$$\frac { 48800 } { x } = \frac { 24000 x } { 210 - x }$$

Solve for x

$$x = \frac{\sqrt{1540921} - 61}{60} \approx 19.67230651$$

$$x=\frac{-\sqrt{1540921}-61}{60}\approx -21.705639843$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,210 since division by zero is not defined. Multiply both sides of the equation by x\left(x-210\right), the least common multiple of x,210-x.

$$\left(x-210\right)\times 48800=-x\times 24000x$$

Use the distributive property to multiply x-210 by 48800.

$$48800x-10248000=-x\times 24000x$$

Multiply x and x to get x^{2}.

$$48800x-10248000=-x^{2}\times 24000$$

Multiply -1 and 24000 to get -24000.

$$48800x-10248000=-24000x^{2}$$

Add 24000x^{2} to both sides.

$$48800x-10248000+24000x^{2}=0$$

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.

$$24000x^{2}+48800x-10248000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 24000 for a, 48800 for b, and -10248000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-48800±\sqrt{48800^{2}-4\times 24000\left(-10248000\right)}}{2\times 24000}$$

Square 48800.

$$x=\frac{-48800±\sqrt{2381440000-4\times 24000\left(-10248000\right)}}{2\times 24000}$$

Multiply -4 times 24000.

$$x=\frac{-48800±\sqrt{2381440000-96000\left(-10248000\right)}}{2\times 24000}$$

Multiply -96000 times -10248000.

$$x=\frac{-48800±\sqrt{2381440000+983808000000}}{2\times 24000}$$

Add 2381440000 to 983808000000.

$$x=\frac{-48800±\sqrt{986189440000}}{2\times 24000}$$

Take the square root of 986189440000.

$$x=\frac{-48800±800\sqrt{1540921}}{2\times 24000}$$

Multiply 2 times 24000.

$$x=\frac{-48800±800\sqrt{1540921}}{48000}$$

Now solve the equation x=\frac{-48800±800\sqrt{1540921}}{48000} when ± is plus. Add -48800 to 800\sqrt{1540921}.

$$x=\frac{800\sqrt{1540921}-48800}{48000}$$

Divide -48800+800\sqrt{1540921} by 48000.

$$x=\frac{\sqrt{1540921}-61}{60}$$

Now solve the equation x=\frac{-48800±800\sqrt{1540921}}{48000} when ± is minus. Subtract 800\sqrt{1540921} from -48800.

$$x=\frac{-800\sqrt{1540921}-48800}{48000}$$

Divide -48800-800\sqrt{1540921} by 48000.

$$x=\frac{-\sqrt{1540921}-61}{60}$$

The equation is now solved.

$$x=\frac{\sqrt{1540921}-61}{60} x=\frac{-\sqrt{1540921}-61}{60}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,210 since division by zero is not defined. Multiply both sides of the equation by x\left(x-210\right), the least common multiple of x,210-x.

$$\left(x-210\right)\times 48800=-x\times 24000x$$

Use the distributive property to multiply x-210 by 48800.

$$48800x-10248000=-x\times 24000x$$

Multiply x and x to get x^{2}.

$$48800x-10248000=-x^{2}\times 24000$$

Multiply -1 and 24000 to get -24000.

$$48800x-10248000=-24000x^{2}$$

Add 24000x^{2} to both sides.

$$48800x-10248000+24000x^{2}=0$$

Add 10248000 to both sides. Anything plus zero gives itself.

$$48800x+24000x^{2}=10248000$$

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.

$$24000x^{2}+48800x=10248000$$

Divide both sides by 24000.

$$\frac{24000x^{2}+48800x}{24000}=\frac{10248000}{24000}$$

Dividing by 24000 undoes the multiplication by 24000.

$$x^{2}+\frac{48800}{24000}x=\frac{10248000}{24000}$$

Reduce the fraction \frac{48800}{24000} to lowest terms by extracting and canceling out 800.

$$x^{2}+\frac{61}{30}x=\frac{10248000}{24000}$$

Divide 10248000 by 24000.

$$x^{2}+\frac{61}{30}x=427$$

Divide \frac{61}{30}, the coefficient of the x term, by 2 to get \frac{61}{60}. Then add the square of \frac{61}{60} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{61}{30}x+\left(\frac{61}{60}\right)^{2}=427+\left(\frac{61}{60}\right)^{2}$$

Square \frac{61}{60} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{61}{30}x+\frac{3721}{3600}=427+\frac{3721}{3600}$$

Add 427 to \frac{3721}{3600}.

$$x^{2}+\frac{61}{30}x+\frac{3721}{3600}=\frac{1540921}{3600}$$

Factor x^{2}+\frac{61}{30}x+\frac{3721}{3600}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{61}{60}\right)^{2}=\frac{1540921}{3600}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{61}{60}\right)^{2}}=\sqrt{\frac{1540921}{3600}}$$

Simplify.

$$x+\frac{61}{60}=\frac{\sqrt{1540921}}{60} x+\frac{61}{60}=-\frac{\sqrt{1540921}}{60}$$

Subtract \frac{61}{60} from both sides of the equation.

$$x=\frac{\sqrt{1540921}-61}{60} x=\frac{-\sqrt{1540921}-61}{60}$$