Solve 3x/x+2-x+1/x-2=-11/5

$$\frac { 3 x } { x + 2 } - \frac { x + 1 } { x - 2 } = - \frac { 11 } { 5 }$$

Solve for x

$$x=-\frac{6}{7}\approx -0.857142857$$

$$x=3$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by 5\left(x-2\right)\left(x+2\right), the least common multiple of x+2,x-2,5.

$$\left(5x-10\right)\times 3x-\left(5x+10\right)\left(x+1\right)=-11\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 5x-10 by 3.

$$\left(15x-30\right)x-\left(5x+10\right)\left(x+1\right)=-11\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 15x-30 by x.

$$15x^{2}-30x-\left(5x+10\right)\left(x+1\right)=-11\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 5x+10 by x+1 and combine like terms.

$$15x^{2}-30x-\left(5x^{2}+15x+10\right)=-11\left(x-2\right)\left(x+2\right)$$

To find the opposite of 5x^{2}+15x+10, find the opposite of each term.

$$15x^{2}-30x-5x^{2}-15x-10=-11\left(x-2\right)\left(x+2\right)$$

Combine 15x^{2} and -5x^{2} to get 10x^{2}.

$$10x^{2}-30x-15x-10=-11\left(x-2\right)\left(x+2\right)$$

Combine -30x and -15x to get -45x.

$$10x^{2}-45x-10=-11\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply -11 by x-2.

$$10x^{2}-45x-10=\left(-11x+22\right)\left(x+2\right)$$

Use the distributive property to multiply -11x+22 by x+2 and combine like terms.

$$10x^{2}-45x-10=-11x^{2}+44$$

Add 11x^{2} to both sides.

$$10x^{2}-45x-10+11x^{2}=44$$

Combine 10x^{2} and 11x^{2} to get 21x^{2}.

$$21x^{2}-45x-10=44$$

Subtract 44 from both sides.

$$21x^{2}-45x-10-44=0$$

Subtract 44 from -10 to get -54.

$$21x^{2}-45x-54=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 21 for a, -45 for b, and -54 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-45\right)±\sqrt{\left(-45\right)^{2}-4\times 21\left(-54\right)}}{2\times 21}$$

Square -45.

$$x=\frac{-\left(-45\right)±\sqrt{2025-4\times 21\left(-54\right)}}{2\times 21}$$

Multiply -4 times 21.

$$x=\frac{-\left(-45\right)±\sqrt{2025-84\left(-54\right)}}{2\times 21}$$

Multiply -84 times -54.

$$x=\frac{-\left(-45\right)±\sqrt{2025+4536}}{2\times 21}$$

Add 2025 to 4536.

$$x=\frac{-\left(-45\right)±\sqrt{6561}}{2\times 21}$$

Take the square root of 6561.

$$x=\frac{-\left(-45\right)±81}{2\times 21}$$

The opposite of -45 is 45.

$$x=\frac{45±81}{2\times 21}$$

Multiply 2 times 21.

$$x=\frac{45±81}{42}$$

Now solve the equation x=\frac{45±81}{42} when ± is plus. Add 45 to 81.

$$x=\frac{126}{42}$$

Divide 126 by 42.

$$x=3$$

Now solve the equation x=\frac{45±81}{42} when ± is minus. Subtract 81 from 45.

$$x=-\frac{36}{42}$$

Reduce the fraction \frac{-36}{42} to lowest terms by extracting and canceling out 6.

$$x=-\frac{6}{7}$$

The equation is now solved.

$$x=3 x=-\frac{6}{7}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by 5\left(x-2\right)\left(x+2\right), the least common multiple of x+2,x-2,5.

$$\left(5x-10\right)\times 3x-\left(5x+10\right)\left(x+1\right)=-11\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 5x-10 by 3.

$$\left(15x-30\right)x-\left(5x+10\right)\left(x+1\right)=-11\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 15x-30 by x.

$$15x^{2}-30x-\left(5x+10\right)\left(x+1\right)=-11\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 5x+10 by x+1 and combine like terms.

$$15x^{2}-30x-\left(5x^{2}+15x+10\right)=-11\left(x-2\right)\left(x+2\right)$$

To find the opposite of 5x^{2}+15x+10, find the opposite of each term.

$$15x^{2}-30x-5x^{2}-15x-10=-11\left(x-2\right)\left(x+2\right)$$

Combine 15x^{2} and -5x^{2} to get 10x^{2}.

$$10x^{2}-30x-15x-10=-11\left(x-2\right)\left(x+2\right)$$

Combine -30x and -15x to get -45x.

$$10x^{2}-45x-10=-11\left(x-2\right)\left(x+2\right)$$

Use the distributive property to multiply -11 by x-2.

$$10x^{2}-45x-10=\left(-11x+22\right)\left(x+2\right)$$

Use the distributive property to multiply -11x+22 by x+2 and combine like terms.

$$10x^{2}-45x-10=-11x^{2}+44$$

Add 11x^{2} to both sides.

$$10x^{2}-45x-10+11x^{2}=44$$

Combine 10x^{2} and 11x^{2} to get 21x^{2}.

$$21x^{2}-45x-10=44$$

Add 10 to both sides.

$$21x^{2}-45x=44+10$$

Add 44 and 10 to get 54.

$$21x^{2}-45x=54$$

Divide both sides by 21.

$$\frac{21x^{2}-45x}{21}=\frac{54}{21}$$

Dividing by 21 undoes the multiplication by 21.

$$x^{2}+\left(-\frac{45}{21}\right)x=\frac{54}{21}$$

Reduce the fraction \frac{-45}{21} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{15}{7}x=\frac{54}{21}$$

Reduce the fraction \frac{54}{21} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{15}{7}x=\frac{18}{7}$$

Divide -\frac{15}{7}, the coefficient of the x term, by 2 to get -\frac{15}{14}. Then add the square of -\frac{15}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{15}{7}x+\left(-\frac{15}{14}\right)^{2}=\frac{18}{7}+\left(-\frac{15}{14}\right)^{2}$$

Square -\frac{15}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{15}{7}x+\frac{225}{196}=\frac{18}{7}+\frac{225}{196}$$

Add \frac{18}{7} to \frac{225}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{15}{7}x+\frac{225}{196}=\frac{729}{196}$$

Factor x^{2}-\frac{15}{7}x+\frac{225}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{15}{14}\right)^{2}=\frac{729}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{14}\right)^{2}}=\sqrt{\frac{729}{196}}$$

Simplify.

$$x-\frac{15}{14}=\frac{27}{14} x-\frac{15}{14}=-\frac{27}{14}$$

Add \frac{15}{14} to both sides of the equation.

$$x=3 x=-\frac{6}{7}$$