Solve 6/x+2=7+10/x-3

$$\frac { 6 } { x + 2 } = 7 + \frac { 10 } { x - 3 }$$

Solve for x

$$x=-\frac{4}{7}\approx -0.571428571$$

$$x=1$$

Steps Using the Quadratic Formula

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

$$\left(x-3\right)\times 6=\left(x-3\right)\left(x+2\right)\times 7+\left(x+2\right)\times 10$$

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

$$6x-18=\left(x-3\right)\left(x+2\right)\times 7+\left(x+2\right)\times 10$$

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

$$6x-18=\left(x^{2}-x-6\right)\times 7+\left(x+2\right)\times 10$$

Use the distributive property to multiply x^{2}-x-6 by 7.

$$6x-18=7x^{2}-7x-42+\left(x+2\right)\times 10$$

Use the distributive property to multiply x+2 by 10.

$$6x-18=7x^{2}-7x-42+10x+20$$

Combine -7x and 10x to get 3x.

$$6x-18=7x^{2}+3x-42+20$$

Add -42 and 20 to get -22.

$$6x-18=7x^{2}+3x-22$$

Subtract 7x^{2} from both sides.

$$6x-18-7x^{2}=3x-22$$

Subtract 3x from both sides.

$$6x-18-7x^{2}-3x=-22$$

Combine 6x and -3x to get 3x.

$$3x-18-7x^{2}=-22$$

Add 22 to both sides.

$$3x-18-7x^{2}+22=0$$

Add -18 and 22 to get 4.

$$3x+4-7x^{2}=0$$

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.

$$-7x^{2}+3x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 3 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-7\right)\times 4}}{2\left(-7\right)}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-7\right)\times 4}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-3±\sqrt{9+28\times 4}}{2\left(-7\right)}$$

Multiply 28 times 4.

$$x=\frac{-3±\sqrt{9+112}}{2\left(-7\right)}$$

Add 9 to 112.

$$x=\frac{-3±\sqrt{121}}{2\left(-7\right)}$$

Take the square root of 121.

$$x=\frac{-3±11}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{-3±11}{-14}$$

Now solve the equation x=\frac{-3±11}{-14} when ± is plus. Add -3 to 11.

$$x=\frac{8}{-14}$$

Reduce the fraction \frac{8}{-14} to lowest terms by extracting and canceling out 2.

$$x=-\frac{4}{7}$$

Now solve the equation x=\frac{-3±11}{-14} when ± is minus. Subtract 11 from -3.

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

Divide -14 by -14.

$$x=1$$

The equation is now solved.

$$x=-\frac{4}{7} x=1$$

Steps for Completing the Square

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

$$\left(x-3\right)\times 6=\left(x-3\right)\left(x+2\right)\times 7+\left(x+2\right)\times 10$$

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

$$6x-18=\left(x-3\right)\left(x+2\right)\times 7+\left(x+2\right)\times 10$$

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

$$6x-18=\left(x^{2}-x-6\right)\times 7+\left(x+2\right)\times 10$$

Use the distributive property to multiply x^{2}-x-6 by 7.

$$6x-18=7x^{2}-7x-42+\left(x+2\right)\times 10$$

Use the distributive property to multiply x+2 by 10.

$$6x-18=7x^{2}-7x-42+10x+20$$

Combine -7x and 10x to get 3x.

$$6x-18=7x^{2}+3x-42+20$$

Add -42 and 20 to get -22.

$$6x-18=7x^{2}+3x-22$$

Subtract 7x^{2} from both sides.

$$6x-18-7x^{2}=3x-22$$

Subtract 3x from both sides.

$$6x-18-7x^{2}-3x=-22$$

Combine 6x and -3x to get 3x.

$$3x-18-7x^{2}=-22$$

Add 18 to both sides.

$$3x-7x^{2}=-22+18$$

Add -22 and 18 to get -4.

$$3x-7x^{2}=-4$$

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.

$$-7x^{2}+3x=-4$$

Divide both sides by -7.

$$\frac{-7x^{2}+3x}{-7}=-\frac{4}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$x^{2}+\frac{3}{-7}x=-\frac{4}{-7}$$

Divide 3 by -7.

$$x^{2}-\frac{3}{7}x=-\frac{4}{-7}$$

Divide -4 by -7.

$$x^{2}-\frac{3}{7}x=\frac{4}{7}$$

Divide -\frac{3}{7}, the coefficient of the x term, by 2 to get -\frac{3}{14}. Then add the square of -\frac{3}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{7}x+\left(-\frac{3}{14}\right)^{2}=\frac{4}{7}+\left(-\frac{3}{14}\right)^{2}$$

Square -\frac{3}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{7}x+\frac{9}{196}=\frac{4}{7}+\frac{9}{196}$$

Add \frac{4}{7} to \frac{9}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3}{7}x+\frac{9}{196}=\frac{121}{196}$$

Factor x^{2}-\frac{3}{7}x+\frac{9}{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}.

$$\left(x-\frac{3}{14}\right)^{2}=\frac{121}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{14}\right)^{2}}=\sqrt{\frac{121}{196}}$$

Simplify.

$$x-\frac{3}{14}=\frac{11}{14} x-\frac{3}{14}=-\frac{11}{14}$$

Add \frac{3}{14} to both sides of the equation.

$$x=1 x=-\frac{4}{7}$$