Solve 490/x+28x-900/x-546=0

$$\frac { 490 } { x } + 28 x - \frac { 900 } { x } - 546 = 0$$

Solve for x

$$x=\frac{\sqrt{86009}}{28}+\frac{39}{4}\approx 20.224032516$$

$$x=-\frac{\sqrt{86009}}{28}+\frac{39}{4}\approx -0.724032516$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$490+28xx-900+x\left(-546\right)=0$$

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

$$490+28x^{2}-900+x\left(-546\right)=0$$

Subtract 900 from 490 to get -410.

$$-410+28x^{2}+x\left(-546\right)=0$$

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.

$$28x^{2}-546x-410=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 28 for a, -546 for b, and -410 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-546\right)±\sqrt{\left(-546\right)^{2}-4\times 28\left(-410\right)}}{2\times 28}$$

Square -546.

$$x=\frac{-\left(-546\right)±\sqrt{298116-4\times 28\left(-410\right)}}{2\times 28}$$

Multiply -4 times 28.

$$x=\frac{-\left(-546\right)±\sqrt{298116-112\left(-410\right)}}{2\times 28}$$

Multiply -112 times -410.

$$x=\frac{-\left(-546\right)±\sqrt{298116+45920}}{2\times 28}$$

Add 298116 to 45920.

$$x=\frac{-\left(-546\right)±\sqrt{344036}}{2\times 28}$$

Take the square root of 344036.

$$x=\frac{-\left(-546\right)±2\sqrt{86009}}{2\times 28}$$

The opposite of -546 is 546.

$$x=\frac{546±2\sqrt{86009}}{2\times 28}$$

Multiply 2 times 28.

$$x=\frac{546±2\sqrt{86009}}{56}$$

Now solve the equation x=\frac{546±2\sqrt{86009}}{56} when ± is plus. Add 546 to 2\sqrt{86009}.

$$x=\frac{2\sqrt{86009}+546}{56}$$

Divide 546+2\sqrt{86009} by 56.

$$x=\frac{\sqrt{86009}}{28}+\frac{39}{4}$$

Now solve the equation x=\frac{546±2\sqrt{86009}}{56} when ± is minus. Subtract 2\sqrt{86009} from 546.

$$x=\frac{546-2\sqrt{86009}}{56}$$

Divide 546-2\sqrt{86009} by 56.

$$x=-\frac{\sqrt{86009}}{28}+\frac{39}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{86009}}{28}+\frac{39}{4} x=-\frac{\sqrt{86009}}{28}+\frac{39}{4}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$490+28xx-900+x\left(-546\right)=0$$

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

$$490+28x^{2}-900+x\left(-546\right)=0$$

Subtract 900 from 490 to get -410.

$$-410+28x^{2}+x\left(-546\right)=0$$

Add 410 to both sides. Anything plus zero gives itself.

$$28x^{2}+x\left(-546\right)=410$$

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.

$$28x^{2}-546x=410$$

Divide both sides by 28.

$$\frac{28x^{2}-546x}{28}=\frac{410}{28}$$

Dividing by 28 undoes the multiplication by 28.

$$x^{2}+\left(-\frac{546}{28}\right)x=\frac{410}{28}$$

Reduce the fraction \frac{-546}{28} to lowest terms by extracting and canceling out 14.

$$x^{2}-\frac{39}{2}x=\frac{410}{28}$$

Reduce the fraction \frac{410}{28} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{39}{2}x=\frac{205}{14}$$

Divide -\frac{39}{2}, the coefficient of the x term, by 2 to get -\frac{39}{4}. Then add the square of -\frac{39}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{39}{2}x+\left(-\frac{39}{4}\right)^{2}=\frac{205}{14}+\left(-\frac{39}{4}\right)^{2}$$

Square -\frac{39}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{39}{2}x+\frac{1521}{16}=\frac{205}{14}+\frac{1521}{16}$$

Add \frac{205}{14} to \frac{1521}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{39}{2}x+\frac{1521}{16}=\frac{12287}{112}$$

Factor x^{2}-\frac{39}{2}x+\frac{1521}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{39}{4}\right)^{2}=\frac{12287}{112}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{39}{4}\right)^{2}}=\sqrt{\frac{12287}{112}}$$

Simplify.

$$x-\frac{39}{4}=\frac{\sqrt{86009}}{28} x-\frac{39}{4}=-\frac{\sqrt{86009}}{28}$$

Add \frac{39}{4} to both sides of the equation.

$$x=\frac{\sqrt{86009}}{28}+\frac{39}{4} x=-\frac{\sqrt{86009}}{28}+\frac{39}{4}$$