Solve 5/x-3-6/x+2=1

$$\frac { 5 } { x - 3 } - \frac { 6 } { x + 2 } = 1$$

Solve for x

$$x=\sqrt{34}\approx 5.830951895$$

$$x=-\sqrt{34}\approx -5.830951895$$

Steps by Finding Square Root

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-3,x+2.

$$\left(x+2\right)\times 5-\left(x-3\right)\times 6=\left(x-3\right)\left(x+2\right)$$

Use the distributive property to multiply x+2 by 5.

$$5x+10-\left(x-3\right)\times 6=\left(x-3\right)\left(x+2\right)$$

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

$$5x+10-\left(6x-18\right)=\left(x-3\right)\left(x+2\right)$$

To find the opposite of 6x-18, find the opposite of each term.

$$5x+10-6x+18=\left(x-3\right)\left(x+2\right)$$

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

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

Add 10 and 18 to get 28.

$$-x+28=\left(x-3\right)\left(x+2\right)$$

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

$$-x+28=x^{2}-x-6$$

Subtract x^{2} from both sides.

$$-x+28-x^{2}=-x-6$$

Add x to both sides.

$$-x+28-x^{2}+x=-6$$

Combine -x and x to get 0.

$$28-x^{2}=-6$$

Subtract 28 from both sides.

$$-x^{2}=-6-28$$

Subtract 28 from -6 to get -34.

$$-x^{2}=-34$$

Divide both sides by -1.

$$x^{2}=\frac{-34}{-1}$$

Fraction \frac{-34}{-1} can be simplified to 34 by removing the negative sign from both the numerator and the denominator.

$$x^{2}=34$$

Take the square root of both sides of the equation.

$$x=\sqrt{34} x=-\sqrt{34}$$

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-3,x+2.

$$\left(x+2\right)\times 5-\left(x-3\right)\times 6=\left(x-3\right)\left(x+2\right)$$

Use the distributive property to multiply x+2 by 5.

$$5x+10-\left(x-3\right)\times 6=\left(x-3\right)\left(x+2\right)$$

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

$$5x+10-\left(6x-18\right)=\left(x-3\right)\left(x+2\right)$$

To find the opposite of 6x-18, find the opposite of each term.

$$5x+10-6x+18=\left(x-3\right)\left(x+2\right)$$

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

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

Add 10 and 18 to get 28.

$$-x+28=\left(x-3\right)\left(x+2\right)$$

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

$$-x+28=x^{2}-x-6$$

Subtract x^{2} from both sides.

$$-x+28-x^{2}=-x-6$$

Add x to both sides.

$$-x+28-x^{2}+x=-6$$

Combine -x and x to get 0.

$$28-x^{2}=-6$$

Add 6 to both sides.

$$28-x^{2}+6=0$$

Add 28 and 6 to get 34.

$$34-x^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$-x^{2}+34=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 0 for b, and 34 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-1\right)\times 34}}{2\left(-1\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-1\right)\times 34}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{0±\sqrt{4\times 34}}{2\left(-1\right)}$$

Multiply 4 times 34.

$$x=\frac{0±\sqrt{136}}{2\left(-1\right)}$$

Take the square root of 136.

$$x=\frac{0±2\sqrt{34}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{0±2\sqrt{34}}{-2}$$

Now solve the equation x=\frac{0±2\sqrt{34}}{-2} when ± is plus.

$$x=-\sqrt{34}$$

Now solve the equation x=\frac{0±2\sqrt{34}}{-2} when ± is minus.

$$x=\sqrt{34}$$

The equation is now solved.

$$x=-\sqrt{34} x=\sqrt{34}$$