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

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

Solve for x (complex solution)

$$x=\sqrt{26}-1\approx 4.099019514$$

$$x=-\left(\sqrt{26}+1\right)\approx -6.099019514$$

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.

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

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

$$2x+4-\left(x-3\right)\times 5=\left(x-3\right)\left(x+2\right)$$

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

$$2x+4-\left(5x-15\right)=\left(x-3\right)\left(x+2\right)$$

To find the opposite of 5x-15, find the opposite of each term.

$$2x+4-5x+15=\left(x-3\right)\left(x+2\right)$$

Combine 2x and -5x to get -3x.

$$-3x+4+15=\left(x-3\right)\left(x+2\right)$$

Add 4 and 15 to get 19.

$$-3x+19=\left(x-3\right)\left(x+2\right)$$

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

$$-3x+19=x^{2}-x-6$$

Subtract x^{2} from both sides.

$$-3x+19-x^{2}=-x-6$$

Add x to both sides.

$$-3x+19-x^{2}+x=-6$$

Combine -3x and x to get -2x.

$$-2x+19-x^{2}=-6$$

Add 6 to both sides.

$$-2x+19-x^{2}+6=0$$

Add 19 and 6 to get 25.

$$-2x+25-x^{2}=0$$

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.

$$-x^{2}-2x+25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -2 for b, and 25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-1\right)\times 25}}{2\left(-1\right)}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-1\right)\times 25}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-2\right)±\sqrt{4+4\times 25}}{2\left(-1\right)}$$

Multiply 4 times 25.

$$x=\frac{-\left(-2\right)±\sqrt{4+100}}{2\left(-1\right)}$$

Add 4 to 100.

$$x=\frac{-\left(-2\right)±\sqrt{104}}{2\left(-1\right)}$$

Take the square root of 104.

$$x=\frac{-\left(-2\right)±2\sqrt{26}}{2\left(-1\right)}$$

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{26}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{2±2\sqrt{26}}{-2}$$

Now solve the equation x=\frac{2±2\sqrt{26}}{-2} when ± is plus. Add 2 to 2\sqrt{26}.

$$x=\frac{2\sqrt{26}+2}{-2}$$

Divide 2+2\sqrt{26} by -2.

$$x=-\left(\sqrt{26}+1\right)$$

Now solve the equation x=\frac{2±2\sqrt{26}}{-2} when ± is minus. Subtract 2\sqrt{26} from 2.

$$x=\frac{2-2\sqrt{26}}{-2}$$

Divide 2-2\sqrt{26} by -2.

$$x=\sqrt{26}-1$$

The equation is now solved.

$$x=-\left(\sqrt{26}+1\right) x=\sqrt{26}-1$$

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-3,x+2.

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

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

$$2x+4-\left(x-3\right)\times 5=\left(x-3\right)\left(x+2\right)$$

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

$$2x+4-\left(5x-15\right)=\left(x-3\right)\left(x+2\right)$$

To find the opposite of 5x-15, find the opposite of each term.

$$2x+4-5x+15=\left(x-3\right)\left(x+2\right)$$

Combine 2x and -5x to get -3x.

$$-3x+4+15=\left(x-3\right)\left(x+2\right)$$

Add 4 and 15 to get 19.

$$-3x+19=\left(x-3\right)\left(x+2\right)$$

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

$$-3x+19=x^{2}-x-6$$

Subtract x^{2} from both sides.

$$-3x+19-x^{2}=-x-6$$

Add x to both sides.

$$-3x+19-x^{2}+x=-6$$

Combine -3x and x to get -2x.

$$-2x+19-x^{2}=-6$$

Subtract 19 from both sides.

$$-2x-x^{2}=-6-19$$

Subtract 19 from -6 to get -25.

$$-2x-x^{2}=-25$$

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.

$$-x^{2}-2x=-25$$

Divide both sides by -1.

$$\frac{-x^{2}-2x}{-1}=-\frac{25}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{2}{-1}\right)x=-\frac{25}{-1}$$

Divide -2 by -1.

$$x^{2}+2x=-\frac{25}{-1}$$

Divide -25 by -1.

$$x^{2}+2x=25$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=25+1^{2}$$

Square 1.

$$x^{2}+2x+1=25+1$$

Add 25 to 1.

$$x^{2}+2x+1=26$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=26$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{26}$$

Simplify.

$$x+1=\sqrt{26} x+1=-\sqrt{26}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{26}-1 x=-\sqrt{26}-1$$

Solve for x

$$x=\sqrt{26}-1\approx 4.099019514$$

$$x=-\sqrt{26}-1\approx -6.099019514$$

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.

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

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

$$2x+4-\left(x-3\right)\times 5=\left(x-3\right)\left(x+2\right)$$

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

$$2x+4-\left(5x-15\right)=\left(x-3\right)\left(x+2\right)$$

To find the opposite of 5x-15, find the opposite of each term.

$$2x+4-5x+15=\left(x-3\right)\left(x+2\right)$$

Combine 2x and -5x to get -3x.

$$-3x+4+15=\left(x-3\right)\left(x+2\right)$$

Add 4 and 15 to get 19.

$$-3x+19=\left(x-3\right)\left(x+2\right)$$

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

$$-3x+19=x^{2}-x-6$$

Subtract x^{2} from both sides.

$$-3x+19-x^{2}=-x-6$$

Add x to both sides.

$$-3x+19-x^{2}+x=-6$$

Combine -3x and x to get -2x.

$$-2x+19-x^{2}=-6$$

Add 6 to both sides.

$$-2x+19-x^{2}+6=0$$

Add 19 and 6 to get 25.

$$-2x+25-x^{2}=0$$

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.

$$-x^{2}-2x+25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -2 for b, and 25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-1\right)\times 25}}{2\left(-1\right)}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-1\right)\times 25}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-2\right)±\sqrt{4+4\times 25}}{2\left(-1\right)}$$

Multiply 4 times 25.

$$x=\frac{-\left(-2\right)±\sqrt{4+100}}{2\left(-1\right)}$$

Add 4 to 100.

$$x=\frac{-\left(-2\right)±\sqrt{104}}{2\left(-1\right)}$$

Take the square root of 104.

$$x=\frac{-\left(-2\right)±2\sqrt{26}}{2\left(-1\right)}$$

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{26}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{2±2\sqrt{26}}{-2}$$

Now solve the equation x=\frac{2±2\sqrt{26}}{-2} when ± is plus. Add 2 to 2\sqrt{26}.

$$x=\frac{2\sqrt{26}+2}{-2}$$

Divide 2+2\sqrt{26} by -2.

$$x=-\left(\sqrt{26}+1\right)$$

Now solve the equation x=\frac{2±2\sqrt{26}}{-2} when ± is minus. Subtract 2\sqrt{26} from 2.

$$x=\frac{2-2\sqrt{26}}{-2}$$

Divide 2-2\sqrt{26} by -2.

$$x=\sqrt{26}-1$$

The equation is now solved.

$$x=-\left(\sqrt{26}+1\right) x=\sqrt{26}-1$$

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-3,x+2.

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

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

$$2x+4-\left(x-3\right)\times 5=\left(x-3\right)\left(x+2\right)$$

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

$$2x+4-\left(5x-15\right)=\left(x-3\right)\left(x+2\right)$$

To find the opposite of 5x-15, find the opposite of each term.

$$2x+4-5x+15=\left(x-3\right)\left(x+2\right)$$

Combine 2x and -5x to get -3x.

$$-3x+4+15=\left(x-3\right)\left(x+2\right)$$

Add 4 and 15 to get 19.

$$-3x+19=\left(x-3\right)\left(x+2\right)$$

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

$$-3x+19=x^{2}-x-6$$

Subtract x^{2} from both sides.

$$-3x+19-x^{2}=-x-6$$

Add x to both sides.

$$-3x+19-x^{2}+x=-6$$

Combine -3x and x to get -2x.

$$-2x+19-x^{2}=-6$$

Subtract 19 from both sides.

$$-2x-x^{2}=-6-19$$

Subtract 19 from -6 to get -25.

$$-2x-x^{2}=-25$$

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.

$$-x^{2}-2x=-25$$

Divide both sides by -1.

$$\frac{-x^{2}-2x}{-1}=-\frac{25}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{2}{-1}\right)x=-\frac{25}{-1}$$

Divide -2 by -1.

$$x^{2}+2x=-\frac{25}{-1}$$

Divide -25 by -1.

$$x^{2}+2x=25$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=25+1^{2}$$

Square 1.

$$x^{2}+2x+1=25+1$$

Add 25 to 1.

$$x^{2}+2x+1=26$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=26$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{26}$$

Simplify.

$$x+1=\sqrt{26} x+1=-\sqrt{26}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{26}-1 x=-\sqrt{26}-1$$