Solve 1/x+1+x+3/(x-2)(x+1)=7x/(x+1)(x-2)-x/(x+1)

$$\frac { 1 } { x + 1 } + \frac { x + 3 } { ( x - 2 ) ( x + 1 ) } = \frac { 7 x } { ( x + 1 ) ( x - 2 ) } - \frac { x } { ( x + 1 ) }$$

Solve for x

$$x = \frac{3 \sqrt{5} + 7}{2} \approx 6.854101966$$

$$x=\frac{7-3\sqrt{5}}{2}\approx 0.145898034$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -1,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+1\right), the least common multiple of x+1,\left(x-2\right)\left(x+1\right).

$$x-2+x+3=7x-\left(x-2\right)x$$

Combine x and x to get 2x.

$$2x-2+3=7x-\left(x-2\right)x$$

Add -2 and 3 to get 1.

$$2x+1=7x-\left(x-2\right)x$$

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

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

To find the opposite of x^{2}-2x, find the opposite of each term.

$$2x+1=7x-x^{2}+2x$$

Combine 7x and 2x to get 9x.

$$2x+1=9x-x^{2}$$

Subtract 9x from both sides.

$$2x+1-9x=-x^{2}$$

Combine 2x and -9x to get -7x.

$$-7x+1=-x^{2}$$

Add x^{2} to both sides.

$$-7x+1+x^{2}=0$$

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.

$$x^{2}-7x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -7 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4}}{2}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4}}{2}$$

Add 49 to -4.

$$x=\frac{-\left(-7\right)±\sqrt{45}}{2}$$

Take the square root of 45.

$$x=\frac{-\left(-7\right)±3\sqrt{5}}{2}$$

The opposite of -7 is 7.

$$x=\frac{7±3\sqrt{5}}{2}$$

Now solve the equation x=\frac{7±3\sqrt{5}}{2} when ± is plus. Add 7 to 3\sqrt{5}.

$$x=\frac{3\sqrt{5}+7}{2}$$

Now solve the equation x=\frac{7±3\sqrt{5}}{2} when ± is minus. Subtract 3\sqrt{5} from 7.

$$x=\frac{7-3\sqrt{5}}{2}$$

The equation is now solved.

$$x=\frac{3\sqrt{5}+7}{2} x=\frac{7-3\sqrt{5}}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -1,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+1\right), the least common multiple of x+1,\left(x-2\right)\left(x+1\right).

$$x-2+x+3=7x-\left(x-2\right)x$$

Combine x and x to get 2x.

$$2x-2+3=7x-\left(x-2\right)x$$

Add -2 and 3 to get 1.

$$2x+1=7x-\left(x-2\right)x$$

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

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

To find the opposite of x^{2}-2x, find the opposite of each term.

$$2x+1=7x-x^{2}+2x$$

Combine 7x and 2x to get 9x.

$$2x+1=9x-x^{2}$$

Subtract 9x from both sides.

$$2x+1-9x=-x^{2}$$

Combine 2x and -9x to get -7x.

$$-7x+1=-x^{2}$$

Add x^{2} to both sides.

$$-7x+1+x^{2}=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$-7x+x^{2}=-1$$

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.

$$x^{2}-7x=-1$$

Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-7x+\frac{49}{4}=-1+\frac{49}{4}$$

Add -1 to \frac{49}{4}.

$$x^{2}-7x+\frac{49}{4}=\frac{45}{4}$$

Factor x^{2}-7x+\frac{49}{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}.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{45}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{45}{4}}$$

Simplify.

$$x-\frac{7}{2}=\frac{3\sqrt{5}}{2} x-\frac{7}{2}=-\frac{3\sqrt{5}}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$x=\frac{3\sqrt{5}+7}{2} x=\frac{7-3\sqrt{5}}{2}$$