To solve the equation [tex]\(\frac{5}{x+2} + \frac{3}{x-3} = 2\)[/tex], we will go through the following steps:
1. Combine the fractions on the left-hand side.
2. Clear the denominators by multiplying through by the common denominator.
3. Simplify the resulting equation.
4. Solve for [tex]\(x\)[/tex].
5. Check the solutions in the original equation to ensure they are valid.
Let's begin:
### Step 1: Combine the fractions
We need a common denominator to combine the fractions on the left-hand side. The common denominator of [tex]\(x+2\)[/tex] and [tex]\(x-3\)[/tex] is [tex]\((x+2)(x-3)\)[/tex]. Rewriting the fractions with this common denominator, we have:
[tex]\[
\frac{5(x-3)}{(x+2)(x-3)} + \frac{3(x+2)}{(x+2)(x-3)}.
\][/tex]
Combining these fractions, we get:
[tex]\[
\frac{5(x-3) + 3(x+2)}{(x+2)(x-3)} = 2.
\][/tex]
### Step 2: Clear the denominators
Multiply both sides of the equation by the common denominator [tex]\((x+2)(x-3)\)[/tex] to clear the fractions:
[tex]\[
5(x-3) + 3(x+2) = 2(x+2)(x-3).
\][/tex]
### Step 3: Simplify
Expand both sides of the equation:
[tex]\[
5x - 15 + 3x + 6 = 2(x^2 - x - 6).
\][/tex]
Combine like terms on the left-hand side:
[tex]\[
8x - 9 = 2x^2 - 2x - 12.
\][/tex]
Move all terms to one side to set the equation to zero:
[tex]\[
2x^2 - 2x - 12 - 8x + 9 = 0.
\][/tex]
Simplify further:
[tex]\[
2x^2 - 10x - 3 = 0.
\][/tex]
### Step 4: Solve for [tex]\(x\)[/tex]
We have a quadratic equation [tex]\(2x^2 - 10x - 3 = 0\)[/tex].
Now, we can use the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 2\)[/tex], [tex]\(b = -10\)[/tex], and [tex]\(c = -3\)[/tex]:
[tex]\[
x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(2)(-3)}}{2(2)}.
\][/tex]
Simplifying inside the square root:
[tex]\[
x = \frac{10 \pm \sqrt{100 + 24}}{4},
\][/tex]
[tex]\[
x = \frac{10 \pm \sqrt{124}}{4},
\][/tex]
[tex]\[
x = \frac{10 \pm 2\sqrt{31}}{4},
\][/tex]
[tex]\[
x = \frac{5 \pm \sqrt{31}}{2}.
\][/tex]
So, the solutions to the equation are:
[tex]\[
x = \frac{5 - \sqrt{31}}{2} \quad \text{and} \quad x = \frac{5 + \sqrt{31}}{2}.
\][/tex]
### Step 5: Check the solutions
Finally, we should check these solutions in the original equation to verify their validity. Upon substitution, both these values satisfy the original equation [tex]\(\frac{5}{x+2} + \frac{3}{x-3} = 2\)[/tex].
Therefore, the solutions to the equation [tex]\(\frac{5}{x+2} + \frac{3}{x-3} = 2\)[/tex] are:
[tex]\[
\boxed{\frac{5 - \sqrt{31}}{2}} \quad \text{and} \quad \boxed{\frac{5 + \sqrt{31}}{2}}.
\][/tex]
