Solve 38-3x/4+x=48/x-1

$$\frac{ 38-3x }{ 4+x } = \frac{ 48 }{ x } -1$$

Solve for x (complex solution)

$$x=\frac{-5\sqrt{15}i-3}{2}\approx -1.5-9.682458366i$$

$$x=\frac{-3+5\sqrt{15}i}{2}\approx -1.5+9.682458366i$$

Steps Using the Quadratic Formula

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

$$x\left(38-3x\right)=\left(x+4\right)\times 48+x\left(x+4\right)\left(-1\right)$$

Use the distributive property to multiply x by 38-3x.

$$38x-3x^{2}=\left(x+4\right)\times 48+x\left(x+4\right)\left(-1\right)$$

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

$$38x-3x^{2}=48x+192+x\left(x+4\right)\left(-1\right)$$

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

$$38x-3x^{2}=48x+192+\left(x^{2}+4x\right)\left(-1\right)$$

Use the distributive property to multiply x^{2}+4x by -1.

$$38x-3x^{2}=48x+192-x^{2}-4x$$

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

$$38x-3x^{2}=44x+192-x^{2}$$

Subtract 44x from both sides.

$$38x-3x^{2}-44x=192-x^{2}$$

Combine 38x and -44x to get -6x.

$$-6x-3x^{2}=192-x^{2}$$

Subtract 192 from both sides.

$$-6x-3x^{2}-192=-x^{2}$$

Add x^{2} to both sides.

$$-6x-3x^{2}-192+x^{2}=0$$

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

$$-6x-2x^{2}-192=0$$

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.

$$-2x^{2}-6x-192=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, -6 for b, and -192 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-2\right)\left(-192\right)}}{2\left(-2\right)}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-2\right)\left(-192\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-\left(-6\right)±\sqrt{36+8\left(-192\right)}}{2\left(-2\right)}$$

Multiply 8 times -192.

$$x=\frac{-\left(-6\right)±\sqrt{36-1536}}{2\left(-2\right)}$$

Add 36 to -1536.

$$x=\frac{-\left(-6\right)±\sqrt{-1500}}{2\left(-2\right)}$$

Take the square root of -1500.

$$x=\frac{-\left(-6\right)±10\sqrt{15}i}{2\left(-2\right)}$$

The opposite of -6 is 6.

$$x=\frac{6±10\sqrt{15}i}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{6±10\sqrt{15}i}{-4}$$

Now solve the equation x=\frac{6±10\sqrt{15}i}{-4} when ± is plus. Add 6 to 10i\sqrt{15}.

$$x=\frac{6+10\sqrt{15}i}{-4}$$

Divide 6+10i\sqrt{15} by -4.

$$x=\frac{-5\sqrt{15}i-3}{2}$$

Now solve the equation x=\frac{6±10\sqrt{15}i}{-4} when ± is minus. Subtract 10i\sqrt{15} from 6.

$$x=\frac{-10\sqrt{15}i+6}{-4}$$

Divide 6-10i\sqrt{15} by -4.

$$x=\frac{-3+5\sqrt{15}i}{2}$$

The equation is now solved.

$$x=\frac{-5\sqrt{15}i-3}{2} x=\frac{-3+5\sqrt{15}i}{2}$$

Steps for Completing the Square

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

$$x\left(38-3x\right)=\left(x+4\right)\times 48+x\left(x+4\right)\left(-1\right)$$

Use the distributive property to multiply x by 38-3x.

$$38x-3x^{2}=\left(x+4\right)\times 48+x\left(x+4\right)\left(-1\right)$$

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

$$38x-3x^{2}=48x+192+x\left(x+4\right)\left(-1\right)$$

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

$$38x-3x^{2}=48x+192+\left(x^{2}+4x\right)\left(-1\right)$$

Use the distributive property to multiply x^{2}+4x by -1.

$$38x-3x^{2}=48x+192-x^{2}-4x$$

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

$$38x-3x^{2}=44x+192-x^{2}$$

Subtract 44x from both sides.

$$38x-3x^{2}-44x=192-x^{2}$$

Combine 38x and -44x to get -6x.

$$-6x-3x^{2}=192-x^{2}$$

Add x^{2} to both sides.

$$-6x-3x^{2}+x^{2}=192$$

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

$$-6x-2x^{2}=192$$

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.

$$-2x^{2}-6x=192$$

Divide both sides by -2.

$$\frac{-2x^{2}-6x}{-2}=\frac{192}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\left(-\frac{6}{-2}\right)x=\frac{192}{-2}$$

Divide -6 by -2.

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

Divide 192 by -2.

$$x^{2}+3x=-96$$

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

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=-96+\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=-96+\frac{9}{4}$$

Add -96 to \frac{9}{4}.

$$x^{2}+3x+\frac{9}{4}=-\frac{375}{4}$$

Factor x^{2}+3x+\frac{9}{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}.

$$\left(x+\frac{3}{2}\right)^{2}=-\frac{375}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{-\frac{375}{4}}$$

Simplify.

$$x+\frac{3}{2}=\frac{5\sqrt{15}i}{2} x+\frac{3}{2}=-\frac{5\sqrt{15}i}{2}$$

Subtract \frac{3}{2} from both sides of the equation.

$$x=\frac{-3+5\sqrt{15}i}{2} x=\frac{-5\sqrt{15}i-3}{2}$$