Question

problem 2: complete the area model
now identify factor pairs of $36x^{2}$ that add to $-13x$.
use the factor pairs for the areas of rectangles ii and iii to continue labeling the area model.
$2x^{2}-13x + 18=$________________

Explanation:

Step1: Factor the quadratic expression

We need to factor \(2x^{2}-13x + 18\).
We look for two numbers \(a\) and \(b\) such that \(a\times b=2\times18 = 36\) and \(a + b=-13\). The numbers are \(-4\) and \(-9\) since \((-4)\times(-9)=36\) and \(-4+(-9)=-13\).
We rewrite the middle - term: \(2x^{2}-4x-9x + 18\).
Group the terms: \((2x^{2}-4x)+(-9x + 18)\).
Factor out the common factors from each group: \(2x(x - 2)-9(x - 2)\).
Then, by the distributive property \(a\times c+b\times c=(a + b)\times c\) (here \(a = 2x\), \(b=-9\), \(c=x - 2\)), we get \((2x-9)(x - 2)\).

Step2: Determine the expression for rectangle III

If we consider the area model for \((2x-9)(x - 2)=2x\times x+2x\times(- 2)+(-9)\times x+(-9)\times(-2)\)
The area of rectangle III is \(-4x\) (from \(2x\times(-2)\))

Answer:

\((2x - 9)(x - 2)\) and the expression for rectangle III is \(-4x\)