Solve 3x/+4-5-x/x+1=2

$$\frac { 3 x } { + 4 } - \frac { 5 - x } { x + 1 } = 2$$

Solve for x

$$x = \frac{\sqrt{337} + 1}{6} \approx 3.226259958$$

$$x=\frac{1-\sqrt{337}}{6}\approx -2.892926625$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by 4\left(x+1\right), the least common multiple of 4,x+1.

$$\left(x+1\right)\times 3x-4\left(5-x\right)=8\left(x+1\right)$$

Use the distributive property to multiply x+1 by 3.

$$\left(3x+3\right)x-4\left(5-x\right)=8\left(x+1\right)$$

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

$$3x^{2}+3x-4\left(5-x\right)=8\left(x+1\right)$$

Use the distributive property to multiply -4 by 5-x.

$$3x^{2}+3x-20+4x=8\left(x+1\right)$$

Combine 3x and 4x to get 7x.

$$3x^{2}+7x-20=8\left(x+1\right)$$

Use the distributive property to multiply 8 by x+1.

$$3x^{2}+7x-20=8x+8$$

Subtract 8x from both sides.

$$3x^{2}+7x-20-8x=8$$

Combine 7x and -8x to get -x.

$$3x^{2}-x-20=8$$

Subtract 8 from both sides.

$$3x^{2}-x-20-8=0$$

Subtract 8 from -20 to get -28.

$$3x^{2}-x-28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -1 for b, and -28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 3\left(-28\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-1\right)±\sqrt{1-12\left(-28\right)}}{2\times 3}$$

Multiply -12 times -28.

$$x=\frac{-\left(-1\right)±\sqrt{1+336}}{2\times 3}$$

Add 1 to 336.

$$x=\frac{-\left(-1\right)±\sqrt{337}}{2\times 3}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{337}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{1±\sqrt{337}}{6}$$

Now solve the equation x=\frac{1±\sqrt{337}}{6} when ± is plus. Add 1 to \sqrt{337}.

$$x=\frac{\sqrt{337}+1}{6}$$

Now solve the equation x=\frac{1±\sqrt{337}}{6} when ± is minus. Subtract \sqrt{337} from 1.

$$x=\frac{1-\sqrt{337}}{6}$$

The equation is now solved.

$$x=\frac{\sqrt{337}+1}{6} x=\frac{1-\sqrt{337}}{6}$$

Steps for Completing the Square

Variable x cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by 4\left(x+1\right), the least common multiple of 4,x+1.

$$\left(x+1\right)\times 3x-4\left(5-x\right)=8\left(x+1\right)$$

Use the distributive property to multiply x+1 by 3.

$$\left(3x+3\right)x-4\left(5-x\right)=8\left(x+1\right)$$

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

$$3x^{2}+3x-4\left(5-x\right)=8\left(x+1\right)$$

Use the distributive property to multiply -4 by 5-x.

$$3x^{2}+3x-20+4x=8\left(x+1\right)$$

Combine 3x and 4x to get 7x.

$$3x^{2}+7x-20=8\left(x+1\right)$$

Use the distributive property to multiply 8 by x+1.

$$3x^{2}+7x-20=8x+8$$

Subtract 8x from both sides.

$$3x^{2}+7x-20-8x=8$$

Combine 7x and -8x to get -x.

$$3x^{2}-x-20=8$$

Add 20 to both sides.

$$3x^{2}-x=8+20$$

Add 8 and 20 to get 28.

$$3x^{2}-x=28$$

Divide both sides by 3.

$$\frac{3x^{2}-x}{3}=\frac{28}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}-\frac{1}{3}x=\frac{28}{3}$$

Divide -\frac{1}{3}, the coefficient of the x term, by 2 to get -\frac{1}{6}. Then add the square of -\frac{1}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{3}x+\left(-\frac{1}{6}\right)^{2}=\frac{28}{3}+\left(-\frac{1}{6}\right)^{2}$$

Square -\frac{1}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{3}x+\frac{1}{36}=\frac{28}{3}+\frac{1}{36}$$

Add \frac{28}{3} to \frac{1}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1}{3}x+\frac{1}{36}=\frac{337}{36}$$

Factor x^{2}-\frac{1}{3}x+\frac{1}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{6}\right)^{2}=\frac{337}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{6}\right)^{2}}=\sqrt{\frac{337}{36}}$$

Simplify.

$$x-\frac{1}{6}=\frac{\sqrt{337}}{6} x-\frac{1}{6}=-\frac{\sqrt{337}}{6}$$

Add \frac{1}{6} to both sides of the equation.

$$x=\frac{\sqrt{337}+1}{6} x=\frac{1-\sqrt{337}}{6}$$