Question

1. factor the following expressions. (fill in the blank to make a true statement) a. 2x^{2}+9x + 4=(__)(__) b. 4x^{2}+ 9. decide if each pair of triangles below is similar. if the triangles are similar by stating the similarity condition you used. also describe a possible transformation that would map one onto the other. if the triangles are not similar, explain why. a. triangle diagram b. equilateral triangles

Explanation:

Step1: Factor the quadratic expression \(2x^{2}+9x + 4\)

For a quadratic expression \(ax^{2}+bx + c\) (here \(a = 2\), \(b=9\), \(c = 4\)), we need to find two numbers \(m\) and \(n\) such that \(m\times n=ac=2\times4 = 8\) and \(m + n=b = 9\). The numbers are \(8\) and \(1\).
We rewrite the middle - term: \(2x^{2}+9x + 4=2x^{2}+8x+x + 4\).

Step2: Group the terms

Group the terms as \((2x^{2}+8x)+(x + 4)\).
Factor out the greatest common factor from each group: \(2x(x + 4)+1(x + 4)\).

Step3: Factor out the common binomial factor

Factor out \((x + 4)\) to get \((2x + 1)(x+4)\).

Answer:

\((2x + 1)(x + 4)\)