Solve 3+58/x-4=48/x+1

$$3+ \frac{ 58 }{ x-4 } = \frac{ 48 }{ x } +1$$

Solve for x (complex solution)

$$x=\frac{-1+\sqrt{383}i}{2}\approx -0.5+9.785192895i$$

$$x=\frac{-\sqrt{383}i-1}{2}\approx -0.5-9.785192895i$$

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.

$$x\left(x-4\right)\times 3+x\times 58=\left(x-4\right)\times 48+x\left(x-4\right)$$

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

$$\left(x^{2}-4x\right)\times 3+x\times 58=\left(x-4\right)\times 48+x\left(x-4\right)$$

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

$$3x^{2}-12x+x\times 58=\left(x-4\right)\times 48+x\left(x-4\right)$$

Combine -12x and x\times 58 to get 46x.

$$3x^{2}+46x=\left(x-4\right)\times 48+x\left(x-4\right)$$

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

$$3x^{2}+46x=48x-192+x\left(x-4\right)$$

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

$$3x^{2}+46x=48x-192+x^{2}-4x$$

Combine 48x and -4x to get 44x.

$$3x^{2}+46x=44x-192+x^{2}$$

Subtract 44x from both sides.

$$3x^{2}+46x-44x=-192+x^{2}$$

Combine 46x and -44x to get 2x.

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

Subtract -192 from both sides.

$$3x^{2}+2x-\left(-192\right)=x^{2}$$

The opposite of -192 is 192.

$$3x^{2}+2x+192=x^{2}$$

Subtract x^{2} from both sides.

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

Combine 3x^{2} and -x^{2} to get 2x^{2}.

$$2x^{2}+2x+192=0$$

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

$$x=\frac{-2±\sqrt{2^{2}-4\times 2\times 192}}{2\times 2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\times 2\times 192}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-2±\sqrt{4-8\times 192}}{2\times 2}$$

Multiply -8 times 192.

$$x=\frac{-2±\sqrt{4-1536}}{2\times 2}$$

Add 4 to -1536.

$$x=\frac{-2±\sqrt{-1532}}{2\times 2}$$

Take the square root of -1532.

$$x=\frac{-2±2\sqrt{383}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-2±2\sqrt{383}i}{4}$$

Now solve the equation x=\frac{-2±2\sqrt{383}i}{4} when ± is plus. Add -2 to 2i\sqrt{383}.

$$x=\frac{-2+2\sqrt{383}i}{4}$$

Divide -2+2i\sqrt{383} by 4.

$$x=\frac{-1+\sqrt{383}i}{2}$$

Now solve the equation x=\frac{-2±2\sqrt{383}i}{4} when ± is minus. Subtract 2i\sqrt{383} from -2.

$$x=\frac{-2\sqrt{383}i-2}{4}$$

Divide -2-2i\sqrt{383} by 4.

$$x=\frac{-\sqrt{383}i-1}{2}$$

The equation is now solved.

$$x=\frac{-1+\sqrt{383}i}{2} x=\frac{-\sqrt{383}i-1}{2}$$

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.

$$x\left(x-4\right)\times 3+x\times 58=\left(x-4\right)\times 48+x\left(x-4\right)$$

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

$$\left(x^{2}-4x\right)\times 3+x\times 58=\left(x-4\right)\times 48+x\left(x-4\right)$$

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

$$3x^{2}-12x+x\times 58=\left(x-4\right)\times 48+x\left(x-4\right)$$

Combine -12x and x\times 58 to get 46x.

$$3x^{2}+46x=\left(x-4\right)\times 48+x\left(x-4\right)$$

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

$$3x^{2}+46x=48x-192+x\left(x-4\right)$$

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

$$3x^{2}+46x=48x-192+x^{2}-4x$$

Combine 48x and -4x to get 44x.

$$3x^{2}+46x=44x-192+x^{2}$$

Subtract 44x from both sides.

$$3x^{2}+46x-44x=-192+x^{2}$$

Combine 46x and -44x to get 2x.

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

Subtract x^{2} from both sides.

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

Combine 3x^{2} and -x^{2} to get 2x^{2}.

$$2x^{2}+2x=-192$$

Divide both sides by 2.

$$\frac{2x^{2}+2x}{2}=-\frac{192}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{2}{2}x=-\frac{192}{2}$$

Divide 2 by 2.

$$x^{2}+x=-\frac{192}{2}$$

Divide -192 by 2.

$$x^{2}+x=-96$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=-96+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=-96+\frac{1}{4}$$

Add -96 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=-\frac{383}{4}$$

Factor x^{2}+x+\frac{1}{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}.

$$\left(x+\frac{1}{2}\right)^{2}=-\frac{383}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{-\frac{383}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{383}i}{2} x+\frac{1}{2}=-\frac{\sqrt{383}i}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{-1+\sqrt{383}i}{2} x=\frac{-\sqrt{383}i-1}{2}$$