Question

problem 4: start the area model
keep working on it.
$x^{2}+12x + 35=$
rectangle expression
i $x^{2}$
ii
iii
iv
product of main diagonal
check expressions

Explanation:

Step1: Factor the quadratic expression

We need to factor \(x^{2}+12x + 35\). We look for two numbers \(a\) and \(b\) such that \(a + b=12\) and \(a\times b = 35\). The numbers \(5\) and \(7\) satisfy these conditions (\(5+7 = 12\) and \(5\times7=35\)).
So, \(x^{2}+12x + 35=(x + 5)(x+7)\)

Step2: Fill in the area - model

For the area - model (rectangle model):

• Rectangle I: \(x\times x=x^{2}\)
• Rectangle II: \(x\times7 = 7x\)
• Rectangle III: \(x\times5=5x\)
• Rectangle IV: \(5\times7 = 35\)
• Product of main diagonal: \(x^{2}\times35=35x^{2}\), and \((5x)\times(7x)=35x^{2}\)

Answer:

\((x + 5)(x + 7)\); Rectangle II: \(7x\); Rectangle III: \(5x\); Rectangle IV: \(35\); Product of Main Diagonal: \(35x^{2}\)