Solve 5/x+3+4/x-2=4

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

Solve for x

$$x=-2$$

$$x = \frac{13}{4} = 3\frac{1}{4} = 3.25$$

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 \left(x-2\right)\left(x+3\right), the least common multiple of x+3,x-2.

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

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

$$5x-10+\left(x+3\right)\times 4=4\left(x-2\right)\left(x+3\right)$$

Use the distributive property to multiply x+3 by 4.

$$5x-10+4x+12=4\left(x-2\right)\left(x+3\right)$$

Combine 5x and 4x to get 9x.

$$9x-10+12=4\left(x-2\right)\left(x+3\right)$$

Add -10 and 12 to get 2.

$$9x+2=4\left(x-2\right)\left(x+3\right)$$

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

$$9x+2=\left(4x-8\right)\left(x+3\right)$$

Use the distributive property to multiply 4x-8 by x+3 and combine like terms.

$$9x+2=4x^{2}+4x-24$$

Subtract 4x^{2} from both sides.

$$9x+2-4x^{2}=4x-24$$

Subtract 4x from both sides.

$$9x+2-4x^{2}-4x=-24$$

Combine 9x and -4x to get 5x.

$$5x+2-4x^{2}=-24$$

Add 24 to both sides.

$$5x+2-4x^{2}+24=0$$

Add 2 and 24 to get 26.

$$5x+26-4x^{2}=0$$

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.

$$-4x^{2}+5x+26=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 5 for b, and 26 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-4\right)\times 26}}{2\left(-4\right)}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-4\right)\times 26}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-5±\sqrt{25+16\times 26}}{2\left(-4\right)}$$

Multiply 16 times 26.

$$x=\frac{-5±\sqrt{25+416}}{2\left(-4\right)}$$

Add 25 to 416.

$$x=\frac{-5±\sqrt{441}}{2\left(-4\right)}$$

Take the square root of 441.

$$x=\frac{-5±21}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-5±21}{-8}$$

Now solve the equation x=\frac{-5±21}{-8} when ± is plus. Add -5 to 21.

$$x=\frac{16}{-8}$$

Divide 16 by -8.

$$x=-2$$

Now solve the equation x=\frac{-5±21}{-8} when ± is minus. Subtract 21 from -5.

$$x=-\frac{26}{-8}$$

Reduce the fraction \frac{-26}{-8} to lowest terms by extracting and canceling out 2.

$$x=\frac{13}{4}$$

The equation is now solved.

$$x=-2 x=\frac{13}{4}$$

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 \left(x-2\right)\left(x+3\right), the least common multiple of x+3,x-2.

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

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

$$5x-10+\left(x+3\right)\times 4=4\left(x-2\right)\left(x+3\right)$$

Use the distributive property to multiply x+3 by 4.

$$5x-10+4x+12=4\left(x-2\right)\left(x+3\right)$$

Combine 5x and 4x to get 9x.

$$9x-10+12=4\left(x-2\right)\left(x+3\right)$$

Add -10 and 12 to get 2.

$$9x+2=4\left(x-2\right)\left(x+3\right)$$

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

$$9x+2=\left(4x-8\right)\left(x+3\right)$$

Use the distributive property to multiply 4x-8 by x+3 and combine like terms.

$$9x+2=4x^{2}+4x-24$$

Subtract 4x^{2} from both sides.

$$9x+2-4x^{2}=4x-24$$

Subtract 4x from both sides.

$$9x+2-4x^{2}-4x=-24$$

Combine 9x and -4x to get 5x.

$$5x+2-4x^{2}=-24$$

Subtract 2 from both sides.

$$5x-4x^{2}=-24-2$$

Subtract 2 from -24 to get -26.

$$5x-4x^{2}=-26$$

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.

$$-4x^{2}+5x=-26$$

Divide both sides by -4.

$$\frac{-4x^{2}+5x}{-4}=-\frac{26}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{5}{-4}x=-\frac{26}{-4}$$

Divide 5 by -4.

$$x^{2}-\frac{5}{4}x=-\frac{26}{-4}$$

Reduce the fraction \frac{-26}{-4} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{5}{4}x=\frac{13}{2}$$

Divide -\frac{5}{4}, the coefficient of the x term, by 2 to get -\frac{5}{8}. Then add the square of -\frac{5}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{4}x+\left(-\frac{5}{8}\right)^{2}=\frac{13}{2}+\left(-\frac{5}{8}\right)^{2}$$

Square -\frac{5}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=\frac{13}{2}+\frac{25}{64}$$

Add \frac{13}{2} to \frac{25}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=\frac{441}{64}$$

Factor x^{2}-\frac{5}{4}x+\frac{25}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{8}\right)^{2}=\frac{441}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{8}\right)^{2}}=\sqrt{\frac{441}{64}}$$

Simplify.

$$x-\frac{5}{8}=\frac{21}{8} x-\frac{5}{8}=-\frac{21}{8}$$

Add \frac{5}{8} to both sides of the equation.

$$x=\frac{13}{4} x=-2$$