Solve 16/x+3-8/1-3x=2

$$\frac{ 16 }{ x+3 } - \frac{ 8 }{ 1-3x } =2$$

Solve for x

$$x=-\frac{1}{3}\approx -0.333333333$$

$$x=7$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,\frac{1}{3} since division by zero is not defined. Multiply both sides of the equation by \left(3x-1\right)\left(x+3\right), the least common multiple of x+3,1-3x.

$$\left(3x-1\right)\times 16-\left(-\left(3+x\right)\times 8\right)=2\left(3x-1\right)\left(x+3\right)$$

Use the distributive property to multiply 3x-1 by 16.

$$48x-16-\left(-\left(3+x\right)\times 8\right)=2\left(3x-1\right)\left(x+3\right)$$

Multiply -1 and 8 to get -8.

$$48x-16-\left(-8\left(3+x\right)\right)=2\left(3x-1\right)\left(x+3\right)$$

Use the distributive property to multiply -8 by 3+x.

$$48x-16-\left(-24-8x\right)=2\left(3x-1\right)\left(x+3\right)$$

To find the opposite of -24-8x, find the opposite of each term.

$$48x-16+24+8x=2\left(3x-1\right)\left(x+3\right)$$

Add -16 and 24 to get 8.

$$48x+8+8x=2\left(3x-1\right)\left(x+3\right)$$

Combine 48x and 8x to get 56x.

$$56x+8=2\left(3x-1\right)\left(x+3\right)$$

Use the distributive property to multiply 2 by 3x-1.

$$56x+8=\left(6x-2\right)\left(x+3\right)$$

Use the distributive property to multiply 6x-2 by x+3 and combine like terms.

$$56x+8=6x^{2}+16x-6$$

Subtract 6x^{2} from both sides.

$$56x+8-6x^{2}=16x-6$$

Subtract 16x from both sides.

$$56x+8-6x^{2}-16x=-6$$

Combine 56x and -16x to get 40x.

$$40x+8-6x^{2}=-6$$

Add 6 to both sides.

$$40x+8-6x^{2}+6=0$$

Add 8 and 6 to get 14.

$$40x+14-6x^{2}=0$$

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.

$$-6x^{2}+40x+14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 40 for b, and 14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-40±\sqrt{40^{2}-4\left(-6\right)\times 14}}{2\left(-6\right)}$$

Square 40.

$$x=\frac{-40±\sqrt{1600-4\left(-6\right)\times 14}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-40±\sqrt{1600+24\times 14}}{2\left(-6\right)}$$

Multiply 24 times 14.

$$x=\frac{-40±\sqrt{1600+336}}{2\left(-6\right)}$$

Add 1600 to 336.

$$x=\frac{-40±\sqrt{1936}}{2\left(-6\right)}$$

Take the square root of 1936.

$$x=\frac{-40±44}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-40±44}{-12}$$

Now solve the equation x=\frac{-40±44}{-12} when ± is plus. Add -40 to 44.

$$x=\frac{4}{-12}$$

Reduce the fraction \frac{4}{-12} to lowest terms by extracting and canceling out 4.

$$x=-\frac{1}{3}$$

Now solve the equation x=\frac{-40±44}{-12} when ± is minus. Subtract 44 from -40.

$$x=-\frac{84}{-12}$$

Divide -84 by -12.

$$x=7$$

The equation is now solved.

$$x=-\frac{1}{3} x=7$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,\frac{1}{3} since division by zero is not defined. Multiply both sides of the equation by \left(3x-1\right)\left(x+3\right), the least common multiple of x+3,1-3x.

$$\left(3x-1\right)\times 16-\left(-\left(3+x\right)\times 8\right)=2\left(3x-1\right)\left(x+3\right)$$

Use the distributive property to multiply 3x-1 by 16.

$$48x-16-\left(-\left(3+x\right)\times 8\right)=2\left(3x-1\right)\left(x+3\right)$$

Multiply -1 and 8 to get -8.

$$48x-16-\left(-8\left(3+x\right)\right)=2\left(3x-1\right)\left(x+3\right)$$

Use the distributive property to multiply -8 by 3+x.

$$48x-16-\left(-24-8x\right)=2\left(3x-1\right)\left(x+3\right)$$

To find the opposite of -24-8x, find the opposite of each term.

$$48x-16+24+8x=2\left(3x-1\right)\left(x+3\right)$$

Add -16 and 24 to get 8.

$$48x+8+8x=2\left(3x-1\right)\left(x+3\right)$$

Combine 48x and 8x to get 56x.

$$56x+8=2\left(3x-1\right)\left(x+3\right)$$

Use the distributive property to multiply 2 by 3x-1.

$$56x+8=\left(6x-2\right)\left(x+3\right)$$

Use the distributive property to multiply 6x-2 by x+3 and combine like terms.

$$56x+8=6x^{2}+16x-6$$

Subtract 6x^{2} from both sides.

$$56x+8-6x^{2}=16x-6$$

Subtract 16x from both sides.

$$56x+8-6x^{2}-16x=-6$$

Combine 56x and -16x to get 40x.

$$40x+8-6x^{2}=-6$$

Subtract 8 from both sides.

$$40x-6x^{2}=-6-8$$

Subtract 8 from -6 to get -14.

$$40x-6x^{2}=-14$$

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.

$$-6x^{2}+40x=-14$$

Divide both sides by -6.

$$\frac{-6x^{2}+40x}{-6}=-\frac{14}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\frac{40}{-6}x=-\frac{14}{-6}$$

Reduce the fraction \frac{40}{-6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{20}{3}x=-\frac{14}{-6}$$

Reduce the fraction \frac{-14}{-6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{20}{3}x=\frac{7}{3}$$

Divide -\frac{20}{3}, the coefficient of the x term, by 2 to get -\frac{10}{3}. Then add the square of -\frac{10}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{20}{3}x+\left(-\frac{10}{3}\right)^{2}=\frac{7}{3}+\left(-\frac{10}{3}\right)^{2}$$

Square -\frac{10}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{20}{3}x+\frac{100}{9}=\frac{7}{3}+\frac{100}{9}$$

Add \frac{7}{3} to \frac{100}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{20}{3}x+\frac{100}{9}=\frac{121}{9}$$

Factor x^{2}-\frac{20}{3}x+\frac{100}{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}.

$$\left(x-\frac{10}{3}\right)^{2}=\frac{121}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{10}{3}\right)^{2}}=\sqrt{\frac{121}{9}}$$

Simplify.

$$x-\frac{10}{3}=\frac{11}{3} x-\frac{10}{3}=-\frac{11}{3}$$

Add \frac{10}{3} to both sides of the equation.

$$x=7 x=-\frac{1}{3}$$