Solve 4/5(x+3)-1/x+2=3/5

$$\frac { 4 } { 5 ( x + 3 ) } - \frac { 1 } { x + 2 } = \frac { 3 } { 5 }$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{11}i-8}{3}\approx -2.666666667-1.105541597i$$

$$x=\frac{-8+\sqrt{11}i}{3}\approx -2.666666667+1.105541597i$$

Steps Using the Quadratic Formula

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

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

Use the distributive property to multiply x+2 by 4.

$$4x+8-\left(5x+15\right)=3\left(x+2\right)\left(x+3\right)$$

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

$$4x+8-5x-15=3\left(x+2\right)\left(x+3\right)$$

Combine 4x and -5x to get -x.

$$-x+8-15=3\left(x+2\right)\left(x+3\right)$$

Subtract 15 from 8 to get -7.

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

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

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

Use the distributive property to multiply 3x+6 by x+3 and combine like terms.

$$-x-7=3x^{2}+15x+18$$

Subtract 3x^{2} from both sides.

$$-x-7-3x^{2}=15x+18$$

Subtract 15x from both sides.

$$-x-7-3x^{2}-15x=18$$

Combine -x and -15x to get -16x.

$$-16x-7-3x^{2}=18$$

Subtract 18 from both sides.

$$-16x-7-3x^{2}-18=0$$

Subtract 18 from -7 to get -25.

$$-16x-25-3x^{2}=0$$

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.

$$-3x^{2}-16x-25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -16 for b, and -25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\left(-3\right)\left(-25\right)}}{2\left(-3\right)}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\left(-3\right)\left(-25\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-16\right)±\sqrt{256+12\left(-25\right)}}{2\left(-3\right)}$$

Multiply 12 times -25.

$$x=\frac{-\left(-16\right)±\sqrt{256-300}}{2\left(-3\right)}$$

Add 256 to -300.

$$x=\frac{-\left(-16\right)±\sqrt{-44}}{2\left(-3\right)}$$

Take the square root of -44.

$$x=\frac{-\left(-16\right)±2\sqrt{11}i}{2\left(-3\right)}$$

The opposite of -16 is 16.

$$x=\frac{16±2\sqrt{11}i}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{16±2\sqrt{11}i}{-6}$$

Now solve the equation x=\frac{16±2\sqrt{11}i}{-6} when ± is plus. Add 16 to 2i\sqrt{11}.

$$x=\frac{16+2\sqrt{11}i}{-6}$$

Divide 16+2i\sqrt{11} by -6.

$$x=\frac{-\sqrt{11}i-8}{3}$$

Now solve the equation x=\frac{16±2\sqrt{11}i}{-6} when ± is minus. Subtract 2i\sqrt{11} from 16.

$$x=\frac{-2\sqrt{11}i+16}{-6}$$

Divide 16-2i\sqrt{11} by -6.

$$x=\frac{-8+\sqrt{11}i}{3}$$

The equation is now solved.

$$x=\frac{-\sqrt{11}i-8}{3} x=\frac{-8+\sqrt{11}i}{3}$$

Steps for Completing the Square

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

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

Use the distributive property to multiply x+2 by 4.

$$4x+8-\left(5x+15\right)=3\left(x+2\right)\left(x+3\right)$$

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

$$4x+8-5x-15=3\left(x+2\right)\left(x+3\right)$$

Combine 4x and -5x to get -x.

$$-x+8-15=3\left(x+2\right)\left(x+3\right)$$

Subtract 15 from 8 to get -7.

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

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

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

Use the distributive property to multiply 3x+6 by x+3 and combine like terms.

$$-x-7=3x^{2}+15x+18$$

Subtract 3x^{2} from both sides.

$$-x-7-3x^{2}=15x+18$$

Subtract 15x from both sides.

$$-x-7-3x^{2}-15x=18$$

Combine -x and -15x to get -16x.

$$-16x-7-3x^{2}=18$$

Add 7 to both sides.

$$-16x-3x^{2}=18+7$$

Add 18 and 7 to get 25.

$$-16x-3x^{2}=25$$

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.

$$-3x^{2}-16x=25$$

Divide both sides by -3.

$$\frac{-3x^{2}-16x}{-3}=\frac{25}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{16}{-3}\right)x=\frac{25}{-3}$$

Divide -16 by -3.

$$x^{2}+\frac{16}{3}x=\frac{25}{-3}$$

Divide 25 by -3.

$$x^{2}+\frac{16}{3}x=-\frac{25}{3}$$

Divide \frac{16}{3}, the coefficient of the x term, by 2 to get \frac{8}{3}. Then add the square of \frac{8}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{16}{3}x+\left(\frac{8}{3}\right)^{2}=-\frac{25}{3}+\left(\frac{8}{3}\right)^{2}$$

Square \frac{8}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{16}{3}x+\frac{64}{9}=-\frac{25}{3}+\frac{64}{9}$$

Add -\frac{25}{3} to \frac{64}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{16}{3}x+\frac{64}{9}=-\frac{11}{9}$$

Factor x^{2}+\frac{16}{3}x+\frac{64}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{8}{3}\right)^{2}=-\frac{11}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{8}{3}\right)^{2}}=\sqrt{-\frac{11}{9}}$$

Simplify.

$$x+\frac{8}{3}=\frac{\sqrt{11}i}{3} x+\frac{8}{3}=-\frac{\sqrt{11}i}{3}$$

Subtract \frac{8}{3} from both sides of the equation.

$$x=\frac{-8+\sqrt{11}i}{3} x=\frac{-\sqrt{11}i-8}{3}$$