Question

use your knowledge of factoring to answer questions 7 and 8.

1. fernando and risa are factoring

5x² - x - 6. they each made a chart below
and selected a pair of integers. which student
is correct? explain your reasoning.
fernando
factors sum
1·-6 -5
-1·6 5
2·-3 -1
risa
factors sum
-3·10 7
3·-10 -7
5·-6 -1

1. lawson is pouring a rectangular concrete

slab for his doghouse. the slab will cover an
area of 5x² + 12x - 9 square feet. write
expressions that represent possible
dimensions for the length and width of the
concrete slab.

Explanation:

Question 7:

Step1: Recall factoring \(ax^2 + bx + c\)

For \(5x^2 - x - 6\), we need two numbers that multiply to \(a\times c=5\times(-6)= - 30\) and add up to \(b=-1\).

Step2: Check Fernando's factors

Fernando's factors: For product \(-30\)? Let's check: \(1\times(-6)=-6
eq - 30\), \(-1\times6 = - 6
eq - 30\), \(2\times(-3)=-6
eq - 30\). So his factor pairs are wrong (they multiply to -6, not -30).

Step3: Check Risa's factors

Risa's factors: \(-3\times10=-30\), sum \(7\); \(3\times(-10)=-30\), sum \(-7\); \(5\times(-6)=-30\), sum \(-1\). Ah, \(5\times(-6)=-30\) and \(5 + (-6)=-1\), which matches \(b = - 1\). So Risa's pair is correct as it multiplies to \(a\times c=-30\) and sums to \(b=-1\).

Step1: Factor \(5x^2+12x - 9\)

We need two numbers that multiply to \(5\times(-9)=-45\) and add to \(12\). Let's find such numbers: \(15\times(-3)=-45\) and \(15+(-3)=12\).

Step2: Rewrite the middle term

Rewrite \(12x\) as \(15x-3x\): \(5x^2+15x - 3x - 9\).

Step3: Group and factor

Group: \((5x^2 + 15x)+(-3x - 9)\). Factor out GCF from each group: \(5x(x + 3)-3(x + 3)\). Then factor out \((x + 3)\): \((5x - 3)(x + 3)\).

Answer:

Risa is correct because her factor pair (\(5\) and \(-6\)) multiplies to \(5\times(-6)=-30\) (which is \(a\times c\) for \(5x^2 - x - 6\)) and adds to \(-1\) (which is \(b\)). Fernando's factors multiply to \(-6\) (not \(-30\)) so they are incorrect.

Question 8: