Question

solve for w.

$2w^2 = 7w + 4$

if there is more than one solution, separate them with commas.
if there is no solution, click on
o solution\.

$w = \square$

Explanation:

Step1: Rearrange the equation

First, we need to rewrite the quadratic equation \(2w^{2}=7w + 4\) in standard form \(ax^{2}+bx + c = 0\). Subtract \(7w\) and \(4\) from both sides:
\(2w^{2}-7w - 4=0\)

Step2: Factor the quadratic (or use quadratic formula)

We can factor the quadratic \(2w^{2}-7w - 4\). We need two numbers that multiply to \(2\times(-4)=-8\) and add up to \(-7\). The numbers are \(-8\) and \(1\).
Rewrite the middle term:
\(2w^{2}-8w+w - 4 = 0\)
Group the terms:
\((2w^{2}-8w)+(w - 4)=0\)
Factor out the common factors from each group:
\(2w(w - 4)+1(w - 4)=0\)
Now, factor out \((w - 4)\):
\((2w + 1)(w - 4)=0\)

Step3: Solve for \(w\)

Using the zero - product property, if \(ab = 0\), then either \(a = 0\) or \(b = 0\).
Case 1: \(2w+1 = 0\)
Subtract \(1\) from both sides: \(2w=-1\)
Divide both sides by \(2\): \(w=-\frac{1}{2}\)
Case 2: \(w - 4=0\)
Add \(4\) to both sides: \(w = 4\)

Answer:

\(-\frac{1}{2},4\)