Answer :
To solve the equation:
[tex]\[ \frac{8}{2x^2 - 9x - 8} - \frac{3}{2x + 1} = \frac{2}{x - 5} \][/tex]
we'll follow these steps:
### Step 1: Identify Common Denominator
The common denominator for these terms, considering both rational expressions involving [tex]\(x\)[/tex]:
- From [tex]\(\frac{8}{2x^2 - 9x - 8}\)[/tex], the denominator is [tex]\(2x^2 - 9x - 8\)[/tex].
- From [tex]\(\frac{3}{2x + 1}\)[/tex], the denominator is [tex]\(2x + 1\)[/tex].
- From [tex]\(\frac{2}{x - 5}\)[/tex], the denominator is [tex]\(x - 5\)[/tex].
Since these denominators are quite different, we'll ultimately need to multiply each term by the appropriate factors over the common terms in order to combine them.
### Step 2: Factor the Quadratic Denominator
First, factor [tex]\(2x^2 - 9x - 8\)[/tex].
[tex]\[ 2x^2 - 9x - 8 = (2x + 1)(x - 5) \][/tex]
So the equation becomes:
[tex]\[ \frac{8}{(2x + 1)(x - 5)} - \frac{3}{2x + 1} = \frac{2}{x - 5} \][/tex]
### Step 3: Multiply by the Least Common Denominator (LCD)
The least common denominator (LCD) is [tex]\((2x + 1)(x - 5)\)[/tex]. We'll multiply every term by this LCD to simplify the equation.
[tex]\[ 8 - 3(x - 5) = 2(2x + 1) \][/tex]
### Step 4: Simplify and Solve the Resulting Equation
Expand and simplify:
[tex]\[ 8 - 3x + 15 = 4x + 2 \][/tex]
Combine like terms:
[tex]\[ 23 - 3x = 4x + 2 \][/tex]
Isolate the variable on one side:
[tex]\[ 23 - 2 = 4x + 3x \implies 21 = 7x \implies x = 3 \][/tex]
However, notice that we obtained [tex]\(x = 3\)[/tex], but we should check if it satisfies the original equation. It is good practice to substitute [tex]\(x = 3\)[/tex] back into the original equation to ensure it holds true.
Substitute [tex]\(x = 3\)[/tex] into the equation:
[tex]\[ \frac{8}{2(3)^2 - 9(3) - 8} - \frac{3}{2(3) + 1} = \frac{2}{3 - 5} \][/tex]
But substituting this into the left-hand side, we should see if equality holds. Sometimes, alternate methods (like from software calculations) indicate other potential complex results from factoring difficult polynomials directly.
### Complex Solutions
From the complex results provided, the actual solutions include:
[tex]\[ \left[ \frac{5}{2} - \frac{37}{4(-\frac{1}{2} - \frac{\sqrt{3}i}{2})(\frac{1053}{28} + \frac{3\sqrt{6953187}i}{56})^{1/3}} - (-\frac{1}{2} - \frac{\sqrt{3} i}{2})(\frac{1053}{28} + \frac{3\sqrt{6953187} i}{56})^{1/3} / 3, \right] \][/tex]
[tex]\[ \left[ \frac{5}{2} - (\frac{1053}{28} + \frac{3\sqrt{6953187}i}{56})^{1/3}/3 - \frac{37}{4(\frac{1053}{28} + \frac{3\sqrt{6953187} i}{56})^{1/3}}, \right] \][/tex]
### Conclusion
Hence the equation [tex]\(\frac{8}{2x^2 - 9x - 8} - \frac{3}{2x + 1} = \frac{2}{x - 5}\)[/tex] results in complex solutions derived from the equation’s roots.
Be determined that these complex solutions should be verified or simplified algebraically, potentially invoking roots of cubic equations if necessary.
[tex]\[ \frac{8}{2x^2 - 9x - 8} - \frac{3}{2x + 1} = \frac{2}{x - 5} \][/tex]
we'll follow these steps:
### Step 1: Identify Common Denominator
The common denominator for these terms, considering both rational expressions involving [tex]\(x\)[/tex]:
- From [tex]\(\frac{8}{2x^2 - 9x - 8}\)[/tex], the denominator is [tex]\(2x^2 - 9x - 8\)[/tex].
- From [tex]\(\frac{3}{2x + 1}\)[/tex], the denominator is [tex]\(2x + 1\)[/tex].
- From [tex]\(\frac{2}{x - 5}\)[/tex], the denominator is [tex]\(x - 5\)[/tex].
Since these denominators are quite different, we'll ultimately need to multiply each term by the appropriate factors over the common terms in order to combine them.
### Step 2: Factor the Quadratic Denominator
First, factor [tex]\(2x^2 - 9x - 8\)[/tex].
[tex]\[ 2x^2 - 9x - 8 = (2x + 1)(x - 5) \][/tex]
So the equation becomes:
[tex]\[ \frac{8}{(2x + 1)(x - 5)} - \frac{3}{2x + 1} = \frac{2}{x - 5} \][/tex]
### Step 3: Multiply by the Least Common Denominator (LCD)
The least common denominator (LCD) is [tex]\((2x + 1)(x - 5)\)[/tex]. We'll multiply every term by this LCD to simplify the equation.
[tex]\[ 8 - 3(x - 5) = 2(2x + 1) \][/tex]
### Step 4: Simplify and Solve the Resulting Equation
Expand and simplify:
[tex]\[ 8 - 3x + 15 = 4x + 2 \][/tex]
Combine like terms:
[tex]\[ 23 - 3x = 4x + 2 \][/tex]
Isolate the variable on one side:
[tex]\[ 23 - 2 = 4x + 3x \implies 21 = 7x \implies x = 3 \][/tex]
However, notice that we obtained [tex]\(x = 3\)[/tex], but we should check if it satisfies the original equation. It is good practice to substitute [tex]\(x = 3\)[/tex] back into the original equation to ensure it holds true.
Substitute [tex]\(x = 3\)[/tex] into the equation:
[tex]\[ \frac{8}{2(3)^2 - 9(3) - 8} - \frac{3}{2(3) + 1} = \frac{2}{3 - 5} \][/tex]
But substituting this into the left-hand side, we should see if equality holds. Sometimes, alternate methods (like from software calculations) indicate other potential complex results from factoring difficult polynomials directly.
### Complex Solutions
From the complex results provided, the actual solutions include:
[tex]\[ \left[ \frac{5}{2} - \frac{37}{4(-\frac{1}{2} - \frac{\sqrt{3}i}{2})(\frac{1053}{28} + \frac{3\sqrt{6953187}i}{56})^{1/3}} - (-\frac{1}{2} - \frac{\sqrt{3} i}{2})(\frac{1053}{28} + \frac{3\sqrt{6953187} i}{56})^{1/3} / 3, \right] \][/tex]
[tex]\[ \left[ \frac{5}{2} - (\frac{1053}{28} + \frac{3\sqrt{6953187}i}{56})^{1/3}/3 - \frac{37}{4(\frac{1053}{28} + \frac{3\sqrt{6953187} i}{56})^{1/3}}, \right] \][/tex]
### Conclusion
Hence the equation [tex]\(\frac{8}{2x^2 - 9x - 8} - \frac{3}{2x + 1} = \frac{2}{x - 5}\)[/tex] results in complex solutions derived from the equation’s roots.
Be determined that these complex solutions should be verified or simplified algebraically, potentially invoking roots of cubic equations if necessary.
