Question

solve by factoring \\(2x^2 + 5x = -2\\)

Explanation:

Step1: Rewrite in standard form

First, we need to rewrite the equation \(2x^{2}+5x = - 2\) in the standard quadratic form \(ax^{2}+bx + c=0\). We add 2 to both sides of the equation:
\(2x^{2}+5x + 2=0\)

Step2: Factor the quadratic

We need to factor the quadratic expression \(2x^{2}+5x + 2\). We look for two numbers that multiply to \(2\times2 = 4\) and add up to 5. The numbers are 1 and 4.
We rewrite the middle term:
\(2x^{2}+x + 4x+2 = 0\)
Group the terms:
\((2x^{2}+x)+(4x + 2)=0\)
Factor out the greatest common factor from each group:
\(x(2x + 1)+2(2x + 1)=0\)
Now, we can factor out the common binomial factor \((2x + 1)\):
\((2x + 1)(x + 2)=0\)

Step3: Solve for x

Using the zero - product property, if \(ab = 0\), then either \(a = 0\) or \(b = 0\).
Set \(2x+1 = 0\):
\(2x=-1\)
\(x=-\frac{1}{2}\)
Set \(x + 2=0\):
\(x=-2\)

Answer:

\(x=-\frac{1}{2}\) or \(x = - 2\)