What is the solution for the equation $\frac{5}{3 b^3-2 b^2-5}=\frac{2}{b^3-2}$?
A. $b=-4$ and $b=0$
B. $b=-4$
C. $b=0$ and $b=4$
D. $b=4$
Answer (2)
Cross-multiply the given equation to eliminate fractions.
Expand and rearrange the equation to obtain a polynomial equation.
Factor the polynomial equation to find the possible values of b .
Check the solutions in the original equation to eliminate extraneous solutions. The solutions are b = 0 and b = 4 .
Explanation
Understanding the Problem We are given the equation 3 b 3 − 2 b 2 − 5 5 = b 3 − 2 2 and asked to find the solution(s) for b .
Cross-Multiplication To solve the equation, we first cross-multiply to eliminate the fractions: 5 ( b 3 − 2 ) = 2 ( 3 b 3 − 2 b 2 − 5 )
Expanding the Equation Next, we expand both sides of the equation: 5 b 3 − 10 = 6 b 3 − 4 b 2 − 10
Rearranging the Equation Now, we rearrange the equation to set it equal to zero: 0 = 6 b 3 − 4 b 2 − 10 − 5 b 3 + 10 0 = b 3 − 4 b 2
Factoring We factor out the common factor b 2 : b 2 ( b − 4 ) = 0
Solving for b Solving for b , we have two possible solutions: b 2 = 0 ⇒ b = 0 b − 4 = 0 ⇒ b = 4 So, b = 0 or b = 4 .
Checking for Extraneous Solutions We need to check for extraneous solutions by plugging the values of b back into the original equation. If b = 0 , the original equation becomes: 3 ( 0 ) 3 − 2 ( 0 ) 2 − 5 5 = ( 0 ) 3 − 2 2 − 5 5 = − 2 2 − 1 = − 1 So b = 0 is a valid solution. If b = 4 , the original equation becomes: 3 ( 4 ) 3 − 2 ( 4 ) 2 − 5 5 = ( 4 ) 3 − 2 2 3 ( 64 ) − 2 ( 16 ) − 5 5 = 64 − 2 2 192 − 32 − 5 5 = 62 2 155 5 = 62 2 31 1 = 31 1 So b = 4 is a valid solution.
Final Answer Therefore, the solutions are b = 0 and b = 4 .
Examples
Understanding how to solve rational equations is crucial in many fields, such as physics and engineering. For example, in electrical engineering, you might use rational equations to analyze circuits involving resistors and capacitors. The values of the components can be related through rational functions, and solving for specific variables helps engineers design and optimize the circuit's performance. Similarly, in fluid dynamics, rational equations can describe the flow rates and pressures in a system of pipes, allowing engineers to predict and control fluid behavior.
The solutions for the equation 3 b 3 − 2 b 2 − 5 5 = b 3 − 2 2 are b = 0 and b = 4 . Therefore, the correct choice is C. b = 0 and b = 4 .
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