Solve 100/x+2-1/4=60/x

$$\frac { 100 } { x + 2 } - \frac { 1 } { 4 } = \frac { 60 } { x }$$

Solve for x

$$x=\sqrt{5761}+79\approx 154.901251637$$

$$x=79-\sqrt{5761}\approx 3.098748363$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by 4x\left(x+2\right), the least common multiple of x+2,4,x.

$$4x\times 100+4x\left(x+2\right)\left(-\frac{1}{4}\right)=\left(4x+8\right)\times 60$$

Multiply 4 and 100 to get 400.

$$400x+4x\left(x+2\right)\left(-\frac{1}{4}\right)=\left(4x+8\right)\times 60$$

Multiply 4 and -\frac{1}{4} to get -1.

$$400x-x\left(x+2\right)=\left(4x+8\right)\times 60$$

Use the distributive property to multiply -x by x+2.

$$400x-x^{2}-2x=\left(4x+8\right)\times 60$$

Combine 400x and -2x to get 398x.

$$398x-x^{2}=\left(4x+8\right)\times 60$$

Use the distributive property to multiply 4x+8 by 60.

$$398x-x^{2}=240x+480$$

Subtract 240x from both sides.

$$398x-x^{2}-240x=480$$

Combine 398x and -240x to get 158x.

$$158x-x^{2}=480$$

Subtract 480 from both sides.

$$158x-x^{2}-480=0$$

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.

$$-x^{2}+158x-480=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 158 for b, and -480 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-158±\sqrt{158^{2}-4\left(-1\right)\left(-480\right)}}{2\left(-1\right)}$$

Square 158.

$$x=\frac{-158±\sqrt{24964-4\left(-1\right)\left(-480\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-158±\sqrt{24964+4\left(-480\right)}}{2\left(-1\right)}$$

Multiply 4 times -480.

$$x=\frac{-158±\sqrt{24964-1920}}{2\left(-1\right)}$$

Add 24964 to -1920.

$$x=\frac{-158±\sqrt{23044}}{2\left(-1\right)}$$

Take the square root of 23044.

$$x=\frac{-158±2\sqrt{5761}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-158±2\sqrt{5761}}{-2}$$

Now solve the equation x=\frac{-158±2\sqrt{5761}}{-2} when ± is plus. Add -158 to 2\sqrt{5761}.

$$x=\frac{2\sqrt{5761}-158}{-2}$$

Divide -158+2\sqrt{5761} by -2.

$$x=79-\sqrt{5761}$$

Now solve the equation x=\frac{-158±2\sqrt{5761}}{-2} when ± is minus. Subtract 2\sqrt{5761} from -158.

$$x=\frac{-2\sqrt{5761}-158}{-2}$$

Divide -158-2\sqrt{5761} by -2.

$$x=\sqrt{5761}+79$$

The equation is now solved.

$$x=79-\sqrt{5761} x=\sqrt{5761}+79$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by 4x\left(x+2\right), the least common multiple of x+2,4,x.

$$4x\times 100+4x\left(x+2\right)\left(-\frac{1}{4}\right)=\left(4x+8\right)\times 60$$

Multiply 4 and 100 to get 400.

$$400x+4x\left(x+2\right)\left(-\frac{1}{4}\right)=\left(4x+8\right)\times 60$$

Multiply 4 and -\frac{1}{4} to get -1.

$$400x-x\left(x+2\right)=\left(4x+8\right)\times 60$$

Use the distributive property to multiply -x by x+2.

$$400x-x^{2}-2x=\left(4x+8\right)\times 60$$

Combine 400x and -2x to get 398x.

$$398x-x^{2}=\left(4x+8\right)\times 60$$

Use the distributive property to multiply 4x+8 by 60.

$$398x-x^{2}=240x+480$$

Subtract 240x from both sides.

$$398x-x^{2}-240x=480$$

Combine 398x and -240x to get 158x.

$$158x-x^{2}=480$$

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.

$$-x^{2}+158x=480$$

Divide both sides by -1.

$$\frac{-x^{2}+158x}{-1}=\frac{480}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{158}{-1}x=\frac{480}{-1}$$

Divide 158 by -1.

$$x^{2}-158x=\frac{480}{-1}$$

Divide 480 by -1.

$$x^{2}-158x=-480$$

Divide -158, the coefficient of the x term, by 2 to get -79. Then add the square of -79 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-158x+\left(-79\right)^{2}=-480+\left(-79\right)^{2}$$

Square -79.

$$x^{2}-158x+6241=-480+6241$$

Add -480 to 6241.

$$x^{2}-158x+6241=5761$$

Factor x^{2}-158x+6241. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-79\right)^{2}=5761$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-79\right)^{2}}=\sqrt{5761}$$

Simplify.

$$x-79=\sqrt{5761} x-79=-\sqrt{5761}$$

Add 79 to both sides of the equation.

$$x=\sqrt{5761}+79 x=79-\sqrt{5761}$$