Let's solve the equation [tex]\(\frac{w}{2w-3} = \frac{4}{w}\)[/tex] step-by-step.
1. Start with the given equation:
[tex]\[
\frac{w}{2w - 3} = \frac{4}{w}
\][/tex]
2. Cross-multiply to eliminate the fractions:
[tex]\[
w \cdot w = 4 \cdot (2w - 3)
\][/tex]
Simplifying, we have:
[tex]\[
w^2 = 8w - 12
\][/tex]
3. Rearrange the equation to bring all terms to one side:
[tex]\[
w^2 - 8w + 12 = 0
\][/tex]
4. Solve the quadratic equation using the quadratic formula where [tex]\(a = 1\)[/tex], [tex]\(b = -8\)[/tex], and [tex]\(c = 12\)[/tex]:
[tex]\[
w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Plugging in the values:
[tex]\[
w = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot 12}}{2 \cdot 1}
\][/tex]
Simplify inside the square root:
[tex]\[
w = \frac{8 \pm \sqrt{64 - 48}}{2}
\][/tex]
[tex]\[
w = \frac{8 \pm \sqrt{16}}{2}
\][/tex]
[tex]\[
w = \frac{8 \pm 4}{2}
\][/tex]
5. Determine the two solutions:
[tex]\[
w = \frac{8 + 4}{2} = \frac{12}{2} = 6
\][/tex]
[tex]\[
w = \frac{8 - 4}{2} = \frac{4}{2} = 2
\][/tex]
So the solutions to the equation [tex]\(\frac{w}{2w-3} = \frac{4}{w}\)[/tex] are [tex]\(w = 2\)[/tex] and [tex]\(w = 6\)[/tex].
Therefore, the correct answer is:
[tex]\[ w = 2 \; \text{and} \; w = 6 \][/tex]
