Solve for $x$.
$\frac{x}{2}=\frac{3 x+4}{2 x}$
A) $x=4,-1$
B) $x=4,1 / 2$
C) $x=4,-1 / 2$
Answer (2)
Multiply both sides by 2 x to get rid of the fractions: x 2 = 3 x + 4 .
Rearrange the equation into the standard quadratic form: x 2 − 3 x − 4 = 0 .
Factor the quadratic equation: ( x − 4 ) ( x + 1 ) = 0 .
Solve for x by setting each factor to zero: x = 4 or x = − 1 . Thus, the solutions are x = 4 , − 1 .
Explanation
Problem Analysis We are given the equation 2 x = 2 x 3 x + 4 and asked to solve for x . This involves algebraic manipulation to isolate x .
Eliminating Fractions First, we multiply both sides of the equation by 2 x to eliminate the fractions: 2 x ⋅ 2 x = 2 x ⋅ 2 x 3 x + 4 This simplifies to: x 2 = 3 x + 4
Rearranging into Quadratic Form Next, we rearrange the equation into a standard quadratic form by subtracting 3 x and 4 from both sides: x 2 − 3 x − 4 = 0
Factoring the Quadratic Now, we factor the quadratic equation: ( x − 4 ) ( x + 1 ) = 0
Solving for x To solve for x , we set each factor equal to zero: x − 4 = 0 or x + 1 = 0 Solving these gives us: x = 4 or x = − 1
Final Answer Therefore, the solutions for x are 4 and − 1 .
Examples
Understanding how to solve equations like this is crucial in many fields, such as physics and engineering, where you often need to find unknown variables based on given relationships. For instance, if you're designing a circuit and know the relationship between voltage and current, you might use similar algebraic techniques to determine the resistance needed for a specific component. Or, in physics, when analyzing motion, you might solve equations to find the time or distance traveled by an object under certain conditions. These skills are fundamental for problem-solving in various scientific and technical contexts.
The solutions for the equation 2 x = 2 x 3 x + 4 are x = 4 and x = − 1 . The correct answer is option A) x = 4 , − 1 .
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