Solve 3x/x+5+1/x-2=7/x^2+3x-10

$$\frac { 3 x } { x + 5 } + \frac { 1 } { x - 2 } = \frac { 7 } { x ^ { 2 } + 3 x - 10 }$$

Solve for x

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

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -5,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+5\right), the least common multiple of x+5,x-2,x^{2}+3x-10.

$$\left(x-2\right)\times 3x+x+5=7$$

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

$$\left(3x-6\right)x+x+5=7$$

Use the distributive property to multiply 3x-6 by x.

$$3x^{2}-6x+x+5=7$$

Combine -6x and x to get -5x.

$$3x^{2}-5x+5=7$$

Subtract 7 from both sides.

$$3x^{2}-5x+5-7=0$$

Subtract 7 from 5 to get -2.

$$3x^{2}-5x-2=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 3x^{2}+ax+bx-2. To find a and b, set up a system to be solved.

$$a+b=-5 ab=3\left(-2\right)=-6$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -6.

$$1,-6 2,-3$$

Calculate the sum for each pair.

$$1-6=-5 2-3=-1$$

The solution is the pair that gives sum -5.

$$a=-6 b=1$$

Rewrite 3x^{2}-5x-2 as \left(3x^{2}-6x\right)+\left(x-2\right).

$$\left(3x^{2}-6x\right)+\left(x-2\right)$$

Factor out 3x in 3x^{2}-6x.

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

Factor out common term x-2 by using distributive property.

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

To find equation solutions, solve x-2=0 and 3x+1=0.

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

Variable x cannot be equal to 2.

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

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -5,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+5\right), the least common multiple of x+5,x-2,x^{2}+3x-10.

$$\left(x-2\right)\times 3x+x+5=7$$

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

$$\left(3x-6\right)x+x+5=7$$

Use the distributive property to multiply 3x-6 by x.

$$3x^{2}-6x+x+5=7$$

Combine -6x and x to get -5x.

$$3x^{2}-5x+5=7$$

Subtract 7 from both sides.

$$3x^{2}-5x+5-7=0$$

Subtract 7 from 5 to get -2.

$$3x^{2}-5x-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -5 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 3\left(-2\right)}}{2\times 3}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 3\left(-2\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-5\right)±\sqrt{25-12\left(-2\right)}}{2\times 3}$$

Multiply -12 times -2.

$$x=\frac{-\left(-5\right)±\sqrt{25+24}}{2\times 3}$$

Add 25 to 24.

$$x=\frac{-\left(-5\right)±\sqrt{49}}{2\times 3}$$

Take the square root of 49.

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

The opposite of -5 is 5.

$$x=\frac{5±7}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{5±7}{6}$$

Now solve the equation x=\frac{5±7}{6} when ± is plus. Add 5 to 7.

$$x=\frac{12}{6}$$

Divide 12 by 6.

$$x=2$$

Now solve the equation x=\frac{5±7}{6} when ± is minus. Subtract 7 from 5.

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

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

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

The equation is now solved.

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

Variable x cannot be equal to 2.

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

Steps for Completing the Square

Variable x cannot be equal to any of the values -5,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+5\right), the least common multiple of x+5,x-2,x^{2}+3x-10.

$$\left(x-2\right)\times 3x+x+5=7$$

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

$$\left(3x-6\right)x+x+5=7$$

Use the distributive property to multiply 3x-6 by x.

$$3x^{2}-6x+x+5=7$$

Combine -6x and x to get -5x.

$$3x^{2}-5x+5=7$$

Subtract 5 from both sides.

$$3x^{2}-5x=7-5$$

Subtract 5 from 7 to get 2.

$$3x^{2}-5x=2$$

Divide both sides by 3.

$$\frac{3x^{2}-5x}{3}=\frac{2}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}-\frac{5}{3}x=\frac{2}{3}$$

Divide -\frac{5}{3}, the coefficient of the x term, by 2 to get -\frac{5}{6}. Then add the square of -\frac{5}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{3}x+\left(-\frac{5}{6}\right)^{2}=\frac{2}{3}+\left(-\frac{5}{6}\right)^{2}$$

Square -\frac{5}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=\frac{2}{3}+\frac{25}{36}$$

Add \frac{2}{3} to \frac{25}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=\frac{49}{36}$$

Factor x^{2}-\frac{5}{3}x+\frac{25}{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}.

$$\left(x-\frac{5}{6}\right)^{2}=\frac{49}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{6}\right)^{2}}=\sqrt{\frac{49}{36}}$$

Simplify.

$$x-\frac{5}{6}=\frac{7}{6} x-\frac{5}{6}=-\frac{7}{6}$$

Add \frac{5}{6} to both sides of the equation.

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

Variable x cannot be equal to 2.

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