QuestionSolve. $\frac{2}{b-2}-\frac{1}{b^{2}-4}=0$

Studdy Solution

STEP 1

1. The equation involves rational expressions with the variable $b$ .
2. We can solve the equation by finding a common denominator and simplifying.
3. The final solution should be a real number.

STEP 2

1. Simplify the expression $\frac{1}{b^2 - 4}$ .
2. Find a common denominator for the terms $\frac{2}{b-2}$ and $\frac{1}{b^2 - 4}$ .
3. Solve the resulting equation for $b$ .
4. Check the solution to ensure it does not make any denominator zero.

STEP 3

Simplify the expression $\frac{1}{b^2 - 4}$ by factoring the denominator.
$b^2 - 4 = (b-2)(b+2)$
So,
$\frac{1}{b^2 - 4} = \frac{1}{(b-2)(b+2)}$

STEP 4

Rewrite the original equation using the simplified form of the second term.
$\frac{2}{b-2} - \frac{1}{(b-2)(b+2)} = 0$
We need a common denominator to combine these fractions.

STEP 5

Identify the common denominator, which is $(b-2)(b+2)$ .
Rewrite each fraction with this common denominator:
$\frac{2(b+2)}{(b-2)(b+2)} - \frac{1}{(b-2)(b+2)} = 0$

STEP 6

Combine the fractions:
$\frac{2(b+2) - 1}{(b-2)(b+2)} = 0$

STEP 7

Set the numerator equal to zero, because a fraction is zero if and only if its numerator is zero (and the denominator is non-zero):
$2(b+2) - 1 = 0$

STEP 8

Solve the equation $2(b+2) - 1 = 0$ for $b$ .
$2b + 4 - 1 = 0$ $2b + 3 = 0$

STEP 9

Isolate $b$ :
$2b = -3$ $b = -\frac{3}{2}$

STEP 10

Check the solution to ensure it does not make any denominator zero.
For $b = -\frac{3}{2}$ : - $b-2 = -\frac{3}{2}-2 = -\frac{7}{2} \neq 0$ - $b+2 = -\frac{3}{2}+2 = \frac{1}{2} \neq 0$
Thus, the solution does not make any denominator zero.
Solution: The solution to the equation is:
$b = -\frac{3}{2}$

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