Question

solve for w.
\\(2w^2 + 5 = -11w\\)
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 to standard quadratic form

We start with the equation \(2w^{2}+5 = - 11w\). To get it into the standard quadratic form \(ax^{2}+bx + c=0\) (here \(x = w\)), we add \(11w\) to both sides of the equation.
\(2w^{2}+11w + 5=0\)

Step2: Factor the quadratic equation

We need to find two numbers that multiply to \(a\times c=2\times5 = 10\) and add up to \(b = 11\). The numbers are \(10\) and \(1\). We rewrite the middle term using these numbers:
\(2w^{2}+10w+w + 5=0\)
Now, we factor by grouping. Group the first two terms and the last two terms:
\((2w^{2}+10w)+(w + 5)=0\)
Factor out the greatest common factor from each group:
\(2w(w + 5)+1(w + 5)=0\)
Now, we can factor out \((w + 5)\):
\((2w + 1)(w+5)=0\)

Step3: Solve for \(w\) using the zero - product property

The zero - product property states that 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: \(w + 5=0\)
Subtract \(5\) from both sides: \(w=-5\)

Answer:

\(-5,-\frac{1}{2}\)