Solve 5x/x^2-9+5/x+3=1-x/3-x

$$\frac { 5 x } { x ^ { 2 } - 9 } + \frac { 5 } { x + 3 } = 1 - \frac { x } { 3 - x }$$

Solve for x

$$x=2$$

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+3\right), the least common multiple of x^{2}-9,x+3,3-x.

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

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

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

Combine 5x and 5x to get 10x.

$$10x-15=\left(x-3\right)\left(x+3\right)-\left(-\left(3+x\right)x\right)$$

Consider \left(x-3\right)\left(x+3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 3.

$$10x-15=x^{2}-9-\left(-\left(3+x\right)x\right)$$

Use the distributive property to multiply -1 by 3+x.

$$10x-15=x^{2}-9-\left(-3-x\right)x$$

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

$$10x-15=x^{2}-9-\left(-3x-x^{2}\right)$$

To find the opposite of -3x-x^{2}, find the opposite of each term.

$$10x-15=x^{2}-9+3x+x^{2}$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$10x-15=2x^{2}-9+3x$$

Subtract 2x^{2} from both sides.

$$10x-15-2x^{2}=-9+3x$$

Subtract -9 from both sides.

$$10x-15-2x^{2}-\left(-9\right)=3x$$

The opposite of -9 is 9.

$$10x-15-2x^{2}+9=3x$$

Subtract 3x from both sides.

$$10x-15-2x^{2}+9-3x=0$$

Add -15 and 9 to get -6.

$$10x-6-2x^{2}-3x=0$$

Combine 10x and -3x to get 7x.

$$7x-6-2x^{2}=0$$

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

$$-2x^{2}+7x-6=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -2x^{2}+ax+bx-6. To find a and b, set up a system to be solved.

$$a+b=7 ab=-2\left(-6\right)=12$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 12.

$$1,12 2,6 3,4$$

Calculate the sum for each pair.

$$1+12=13 2+6=8 3+4=7$$

The solution is the pair that gives sum 7.

$$a=4 b=3$$

Rewrite -2x^{2}+7x-6 as \left(-2x^{2}+4x\right)+\left(3x-6\right).

$$\left(-2x^{2}+4x\right)+\left(3x-6\right)$$

Factor out 2x in the first and -3 in the second group.

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

Factor out common term -x+2 by using distributive property.

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

To find equation solutions, solve -x+2=0 and 2x-3=0.

$$x=2 x=\frac{3}{2}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+3\right), the least common multiple of x^{2}-9,x+3,3-x.

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

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

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

Combine 5x and 5x to get 10x.

$$10x-15=\left(x-3\right)\left(x+3\right)-\left(-\left(3+x\right)x\right)$$

Consider \left(x-3\right)\left(x+3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 3.

$$10x-15=x^{2}-9-\left(-\left(3+x\right)x\right)$$

Use the distributive property to multiply -1 by 3+x.

$$10x-15=x^{2}-9-\left(-3-x\right)x$$

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

$$10x-15=x^{2}-9-\left(-3x-x^{2}\right)$$

To find the opposite of -3x-x^{2}, find the opposite of each term.

$$10x-15=x^{2}-9+3x+x^{2}$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$10x-15=2x^{2}-9+3x$$

Subtract 2x^{2} from both sides.

$$10x-15-2x^{2}=-9+3x$$

Subtract -9 from both sides.

$$10x-15-2x^{2}-\left(-9\right)=3x$$

The opposite of -9 is 9.

$$10x-15-2x^{2}+9=3x$$

Subtract 3x from both sides.

$$10x-15-2x^{2}+9-3x=0$$

Add -15 and 9 to get -6.

$$10x-6-2x^{2}-3x=0$$

Combine 10x and -3x to get 7x.

$$7x-6-2x^{2}=0$$

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.

$$-2x^{2}+7x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 7 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-2\right)\left(-6\right)}}{2\left(-2\right)}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-2\right)\left(-6\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-7±\sqrt{49+8\left(-6\right)}}{2\left(-2\right)}$$

Multiply 8 times -6.

$$x=\frac{-7±\sqrt{49-48}}{2\left(-2\right)}$$

Add 49 to -48.

$$x=\frac{-7±\sqrt{1}}{2\left(-2\right)}$$

Take the square root of 1.

$$x=\frac{-7±1}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-7±1}{-4}$$

Now solve the equation x=\frac{-7±1}{-4} when ± is plus. Add -7 to 1.

$$x=-\frac{6}{-4}$$

Reduce the fraction \frac{-6}{-4} to lowest terms by extracting and canceling out 2.

$$x=\frac{3}{2}$$

Now solve the equation x=\frac{-7±1}{-4} when ± is minus. Subtract 1 from -7.

$$x=-\frac{8}{-4}$$

Divide -8 by -4.

$$x=2$$

The equation is now solved.

$$x=\frac{3}{2} x=2$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+3\right), the least common multiple of x^{2}-9,x+3,3-x.

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

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

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

Combine 5x and 5x to get 10x.

$$10x-15=\left(x-3\right)\left(x+3\right)-\left(-\left(3+x\right)x\right)$$

Consider \left(x-3\right)\left(x+3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 3.

$$10x-15=x^{2}-9-\left(-\left(3+x\right)x\right)$$

Use the distributive property to multiply -1 by 3+x.

$$10x-15=x^{2}-9-\left(-3-x\right)x$$

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

$$10x-15=x^{2}-9-\left(-3x-x^{2}\right)$$

To find the opposite of -3x-x^{2}, find the opposite of each term.

$$10x-15=x^{2}-9+3x+x^{2}$$

Combine x^{2} and x^{2} to get 2x^{2}.

$$10x-15=2x^{2}-9+3x$$

Subtract 2x^{2} from both sides.

$$10x-15-2x^{2}=-9+3x$$

Subtract 3x from both sides.

$$10x-15-2x^{2}-3x=-9$$

Combine 10x and -3x to get 7x.

$$7x-15-2x^{2}=-9$$

Add 15 to both sides.

$$7x-2x^{2}=-9+15$$

Add -9 and 15 to get 6.

$$7x-2x^{2}=6$$

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.

$$-2x^{2}+7x=6$$

Divide both sides by -2.

$$\frac{-2x^{2}+7x}{-2}=\frac{6}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{7}{-2}x=\frac{6}{-2}$$

Divide 7 by -2.

$$x^{2}-\frac{7}{2}x=\frac{6}{-2}$$

Divide 6 by -2.

$$x^{2}-\frac{7}{2}x=-3$$

Divide -\frac{7}{2}, the coefficient of the x term, by 2 to get -\frac{7}{4}. Then add the square of -\frac{7}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{2}x+\left(-\frac{7}{4}\right)^{2}=-3+\left(-\frac{7}{4}\right)^{2}$$

Square -\frac{7}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=-3+\frac{49}{16}$$

Add -3 to \frac{49}{16}.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{1}{16}$$

Factor x^{2}-\frac{7}{2}x+\frac{49}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{4}\right)^{2}=\frac{1}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{4}\right)^{2}}=\sqrt{\frac{1}{16}}$$

Simplify.

$$x-\frac{7}{4}=\frac{1}{4} x-\frac{7}{4}=-\frac{1}{4}$$

Add \frac{7}{4} to both sides of the equation.

$$x=2 x=\frac{3}{2}$$