Solve 84/x-4=1+75/x

$$\frac { 84 } { x - 4 } = 1 + \frac { 75 } { x }$$

Solve for x

$$x=-12$$

$$x=25$$

Steps Using the Quadratic Formula

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

$$x\times 84=x\left(x-4\right)+\left(x-4\right)\times 75$$

Use the distributive property to multiply x by x-4.

$$x\times 84=x^{2}-4x+\left(x-4\right)\times 75$$

Use the distributive property to multiply x-4 by 75.

$$x\times 84=x^{2}-4x+75x-300$$

Combine -4x and 75x to get 71x.

$$x\times 84=x^{2}+71x-300$$

Subtract x^{2} from both sides.

$$x\times 84-x^{2}=71x-300$$

Subtract 71x from both sides.

$$x\times 84-x^{2}-71x=-300$$

Combine x\times 84 and -71x to get 13x.

$$13x-x^{2}=-300$$

Add 300 to both sides.

$$13x-x^{2}+300=0$$

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.

$$-x^{2}+13x+300=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 13 for b, and 300 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-1\right)\times 300}}{2\left(-1\right)}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\left(-1\right)\times 300}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-13±\sqrt{169+4\times 300}}{2\left(-1\right)}$$

Multiply 4 times 300.

$$x=\frac{-13±\sqrt{169+1200}}{2\left(-1\right)}$$

Add 169 to 1200.

$$x=\frac{-13±\sqrt{1369}}{2\left(-1\right)}$$

Take the square root of 1369.

$$x=\frac{-13±37}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-13±37}{-2}$$

Now solve the equation x=\frac{-13±37}{-2} when ± is plus. Add -13 to 37.

$$x=\frac{24}{-2}$$

Divide 24 by -2.

$$x=-12$$

Now solve the equation x=\frac{-13±37}{-2} when ± is minus. Subtract 37 from -13.

$$x=-\frac{50}{-2}$$

Divide -50 by -2.

$$x=25$$

The equation is now solved.

$$x=-12 x=25$$

Steps for Completing the Square

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

$$x\times 84=x\left(x-4\right)+\left(x-4\right)\times 75$$

Use the distributive property to multiply x by x-4.

$$x\times 84=x^{2}-4x+\left(x-4\right)\times 75$$

Use the distributive property to multiply x-4 by 75.

$$x\times 84=x^{2}-4x+75x-300$$

Combine -4x and 75x to get 71x.

$$x\times 84=x^{2}+71x-300$$

Subtract x^{2} from both sides.

$$x\times 84-x^{2}=71x-300$$

Subtract 71x from both sides.

$$x\times 84-x^{2}-71x=-300$$

Combine x\times 84 and -71x to get 13x.

$$13x-x^{2}=-300$$

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.

$$-x^{2}+13x=-300$$

Divide both sides by -1.

$$\frac{-x^{2}+13x}{-1}=-\frac{300}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{13}{-1}x=-\frac{300}{-1}$$

Divide 13 by -1.

$$x^{2}-13x=-\frac{300}{-1}$$

Divide -300 by -1.

$$x^{2}-13x=300$$

Divide -13, the coefficient of the x term, by 2 to get -\frac{13}{2}. Then add the square of -\frac{13}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-13x+\left(-\frac{13}{2}\right)^{2}=300+\left(-\frac{13}{2}\right)^{2}$$

Square -\frac{13}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-13x+\frac{169}{4}=300+\frac{169}{4}$$

Add 300 to \frac{169}{4}.

$$x^{2}-13x+\frac{169}{4}=\frac{1369}{4}$$

Factor x^{2}-13x+\frac{169}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{13}{2}\right)^{2}=\frac{1369}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{2}\right)^{2}}=\sqrt{\frac{1369}{4}}$$

Simplify.

$$x-\frac{13}{2}=\frac{37}{2} x-\frac{13}{2}=-\frac{37}{2}$$

Add \frac{13}{2} to both sides of the equation.

$$x=25 x=-12$$