Solve 3+38/x-4=48/x

$$3 + \frac { 38 } { x - 4 } = \frac { 48 } { x }$$

Solve for x (complex solution)

$$x=\frac{11+\sqrt{455}i}{3}\approx 3.666666667+7.110243003i$$

$$x=\frac{-\sqrt{455}i+11}{3}\approx 3.666666667-7.110243003i$$

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.

$$x\left(x-4\right)\times 3+x\times 38=\left(x-4\right)\times 48$$

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

$$\left(x^{2}-4x\right)\times 3+x\times 38=\left(x-4\right)\times 48$$

Use the distributive property to multiply x^{2}-4x by 3.

$$3x^{2}-12x+x\times 38=\left(x-4\right)\times 48$$

Combine -12x and x\times 38 to get 26x.

$$3x^{2}+26x=\left(x-4\right)\times 48$$

Use the distributive property to multiply x-4 by 48.

$$3x^{2}+26x=48x-192$$

Subtract 48x from both sides.

$$3x^{2}+26x-48x=-192$$

Combine 26x and -48x to get -22x.

$$3x^{2}-22x=-192$$

Add 192 to both sides.

$$3x^{2}-22x+192=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -22 for b, and 192 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\times 3\times 192}}{2\times 3}$$

Square -22.

$$x=\frac{-\left(-22\right)±\sqrt{484-4\times 3\times 192}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-22\right)±\sqrt{484-12\times 192}}{2\times 3}$$

Multiply -12 times 192.

$$x=\frac{-\left(-22\right)±\sqrt{484-2304}}{2\times 3}$$

Add 484 to -2304.

$$x=\frac{-\left(-22\right)±\sqrt{-1820}}{2\times 3}$$

Take the square root of -1820.

$$x=\frac{-\left(-22\right)±2\sqrt{455}i}{2\times 3}$$

The opposite of -22 is 22.

$$x=\frac{22±2\sqrt{455}i}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{22±2\sqrt{455}i}{6}$$

Now solve the equation x=\frac{22±2\sqrt{455}i}{6} when ± is plus. Add 22 to 2i\sqrt{455}.

$$x=\frac{22+2\sqrt{455}i}{6}$$

Divide 22+2i\sqrt{455} by 6.

$$x=\frac{11+\sqrt{455}i}{3}$$

Now solve the equation x=\frac{22±2\sqrt{455}i}{6} when ± is minus. Subtract 2i\sqrt{455} from 22.

$$x=\frac{-2\sqrt{455}i+22}{6}$$

Divide 22-2i\sqrt{455} by 6.

$$x=\frac{-\sqrt{455}i+11}{3}$$

The equation is now solved.

$$x=\frac{11+\sqrt{455}i}{3} x=\frac{-\sqrt{455}i+11}{3}$$

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.

$$x\left(x-4\right)\times 3+x\times 38=\left(x-4\right)\times 48$$

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

$$\left(x^{2}-4x\right)\times 3+x\times 38=\left(x-4\right)\times 48$$

Use the distributive property to multiply x^{2}-4x by 3.

$$3x^{2}-12x+x\times 38=\left(x-4\right)\times 48$$

Combine -12x and x\times 38 to get 26x.

$$3x^{2}+26x=\left(x-4\right)\times 48$$

Use the distributive property to multiply x-4 by 48.

$$3x^{2}+26x=48x-192$$

Subtract 48x from both sides.

$$3x^{2}+26x-48x=-192$$

Combine 26x and -48x to get -22x.

$$3x^{2}-22x=-192$$

Divide both sides by 3.

$$\frac{3x^{2}-22x}{3}=-\frac{192}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}-\frac{22}{3}x=-\frac{192}{3}$$

Divide -192 by 3.

$$x^{2}-\frac{22}{3}x=-64$$

Divide -\frac{22}{3}, the coefficient of the x term, by 2 to get -\frac{11}{3}. Then add the square of -\frac{11}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{22}{3}x+\left(-\frac{11}{3}\right)^{2}=-64+\left(-\frac{11}{3}\right)^{2}$$

Square -\frac{11}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{22}{3}x+\frac{121}{9}=-64+\frac{121}{9}$$

Add -64 to \frac{121}{9}.

$$x^{2}-\frac{22}{3}x+\frac{121}{9}=-\frac{455}{9}$$

Factor x^{2}-\frac{22}{3}x+\frac{121}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{11}{3}\right)^{2}=-\frac{455}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{3}\right)^{2}}=\sqrt{-\frac{455}{9}}$$

Simplify.

$$x-\frac{11}{3}=\frac{\sqrt{455}i}{3} x-\frac{11}{3}=-\frac{\sqrt{455}i}{3}$$

Add \frac{11}{3} to both sides of the equation.

$$x=\frac{11+\sqrt{455}i}{3} x=\frac{-\sqrt{455}i+11}{3}$$