Solve 4/x-2-x-3/x+4=0

$$\frac{ 4 }{ x-2 } - \frac{ x-3 }{ x+4 } =0$$

Solve for x

$$x=-1$$

$$x=10$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -4,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+4\right), the least common multiple of x-2,x+4.

$$\left(x+4\right)\times 4-\left(x-2\right)\left(x-3\right)=0$$

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

$$4x+16-\left(x-2\right)\left(x-3\right)=0$$

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

$$4x+16-\left(x^{2}-5x+6\right)=0$$

To find the opposite of x^{2}-5x+6, find the opposite of each term.

$$4x+16-x^{2}+5x-6=0$$

Combine 4x and 5x to get 9x.

$$9x+16-x^{2}-6=0$$

Subtract 6 from 16 to get 10.

$$9x+10-x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}+9x+10=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+10. To find a and b, set up a system to be solved.

$$a+b=9 ab=-10=-10$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -10.

$$-1,10 -2,5$$

Calculate the sum for each pair.

$$-1+10=9 -2+5=3$$

The solution is the pair that gives sum 9.

$$a=10 b=-1$$

Rewrite -x^{2}+9x+10 as \left(-x^{2}+10x\right)+\left(-x+10\right).

$$\left(-x^{2}+10x\right)+\left(-x+10\right)$$

Factor out -x in the first and -1 in the second group.

$$-x\left(x-10\right)-\left(x-10\right)$$

Factor out common term x-10 by using distributive property.

$$\left(x-10\right)\left(-x-1\right)$$

To find equation solutions, solve x-10=0 and -x-1=0.

$$x=10 x=-1$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+4\right), the least common multiple of x-2,x+4.

$$\left(x+4\right)\times 4-\left(x-2\right)\left(x-3\right)=0$$

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

$$4x+16-\left(x-2\right)\left(x-3\right)=0$$

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

$$4x+16-\left(x^{2}-5x+6\right)=0$$

To find the opposite of x^{2}-5x+6, find the opposite of each term.

$$4x+16-x^{2}+5x-6=0$$

Combine 4x and 5x to get 9x.

$$9x+16-x^{2}-6=0$$

Subtract 6 from 16 to get 10.

$$9x+10-x^{2}=0$$

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.

$$-x^{2}+9x+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 9 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-9±\sqrt{81+4\times 10}}{2\left(-1\right)}$$

Multiply 4 times 10.

$$x=\frac{-9±\sqrt{81+40}}{2\left(-1\right)}$$

Add 81 to 40.

$$x=\frac{-9±\sqrt{121}}{2\left(-1\right)}$$

Take the square root of 121.

$$x=\frac{-9±11}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-9±11}{-2}$$

Now solve the equation x=\frac{-9±11}{-2} when ± is plus. Add -9 to 11.

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

Divide 2 by -2.

$$x=-1$$

Now solve the equation x=\frac{-9±11}{-2} when ± is minus. Subtract 11 from -9.

$$x=-\frac{20}{-2}$$

Divide -20 by -2.

$$x=10$$

The equation is now solved.

$$x=-1 x=10$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+4\right), the least common multiple of x-2,x+4.

$$\left(x+4\right)\times 4-\left(x-2\right)\left(x-3\right)=0$$

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

$$4x+16-\left(x-2\right)\left(x-3\right)=0$$

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

$$4x+16-\left(x^{2}-5x+6\right)=0$$

To find the opposite of x^{2}-5x+6, find the opposite of each term.

$$4x+16-x^{2}+5x-6=0$$

Combine 4x and 5x to get 9x.

$$9x+16-x^{2}-6=0$$

Subtract 6 from 16 to get 10.

$$9x+10-x^{2}=0$$

Subtract 10 from both sides. Anything subtracted from zero gives its negation.

$$9x-x^{2}=-10$$

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.

$$-x^{2}+9x=-10$$

Divide both sides by -1.

$$\frac{-x^{2}+9x}{-1}=-\frac{10}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{9}{-1}x=-\frac{10}{-1}$$

Divide 9 by -1.

$$x^{2}-9x=-\frac{10}{-1}$$

Divide -10 by -1.

$$x^{2}-9x=10$$

Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=10+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=10+\frac{81}{4}$$

Add 10 to \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{121}{4}$$

Factor x^{2}-9x+\frac{81}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Simplify.

$$x-\frac{9}{2}=\frac{11}{2} x-\frac{9}{2}=-\frac{11}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$x=10 x=-1$$