Question
Enter the values for \( a \) and \( b \) that complete the sum: \( \frac{3}{x}+\frac{5}{x^{2}}=\frac{a x+b}{x^{2}} \)
Ask by Wright Graham. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( a = 3 \) and \( b = 5 \).
Solution
We start with the equation:
\[
\frac{3}{x}+\frac{5}{x^{2}}=\frac{a x+b}{x^{2}}
\]
**Step 1.** Express the left-hand side with a common denominator. The common denominator is \( x^2 \), so we write:
\[
\frac{3}{x} = \frac{3x}{x^{2}}
\]
Thus:
\[
\frac{3x}{x^{2}}+\frac{5}{x^{2}}=\frac{3x+5}{x^{2}}
\]
**Step 2.** Set the numerators equal since the denominators are the same:
\[
3x+5 = ax+b
\]
**Step 3.** Equating coefficients for like terms, we have:
For the coefficient of \( x \):
\[
3 = a
\]
For the constant term:
\[
5 = b
\]
Therefore, the values are:
\[
a=3, \quad b=5.
\]
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Simplify this solution The Deep Dive
To combine the fractions on the left side, we should rewrite both fractions with a common denominator. The common denominator here is \( x^2 \). The first term, \( \frac{3}{x} \), can be rewritten as: \[ \frac{3}{x} = \frac{3x}{x^2} \] Now, we combine it with the second term: \[ \frac{3x}{x^2} + \frac{5}{x^2} = \frac{3x + 5}{x^2} \] Now, we can compare this with \( \frac{a x + b}{x^{2}} \), leading us to conclude that: - \( a = 3 \) - \( b = 5 \) So the values for \( a \) and \( b \) that complete the sum are \( a = 3 \) and \( b = 5 \).
