![p^2+2(7)p+7^2=(p+7)^2](https://tex.z-dn.net/?f=p%5E2%2B2%287%29p%2B7%5E2%3D%28p%2B7%29%5E2)
Justification:
Given: ![p^2+14p+49](https://tex.z-dn.net/?f=p%5E2%2B14p%2B49)
Split the middle term;
![=p^2+7p+7p+49](https://tex.z-dn.net/?f=%3Dp%5E2%2B7p%2B7p%2B49)
Factor:
![=p(p+7)+7(p+7)](https://tex.z-dn.net/?f=%3Dp%28p%2B7%29%2B7%28p%2B7%29)
![=(p+7)(p+7)](https://tex.z-dn.net/?f=%3D%28p%2B7%29%28p%2B7%29)
![=(p+7)^2](https://tex.z-dn.net/?f=%3D%28p%2B7%29%5E2)
QUESTION 3a
The given expression for the area of the rectangle is
.
This is equal to the indicated area which is ![289\;in^2](https://tex.z-dn.net/?f=289%5C%3Bin%5E2)
This implies that
![36x^2-12x+1=289](https://tex.z-dn.net/?f=36x%5E2-12x%2B1%3D289)
![\Rightarrow (6x-1)^2=289](https://tex.z-dn.net/?f=%5CRightarrow%20%286x-1%29%5E2%3D289)
![\Rightarrow (6x-1)(6x-1)=289](https://tex.z-dn.net/?f=%5CRightarrow%20%286x-1%29%286x-1%29%3D289)
![\Rightarrow l=(6x-1),w=(6x-1)](https://tex.z-dn.net/?f=%5CRightarrow%20l%3D%286x-1%29%2Cw%3D%286x-1%29)
This implies that, the dimensions of the rectangle are equal;
Using the square root method, we obtain
![\Rightarrow (6x-1)=\pm \sqrt{289}](https://tex.z-dn.net/?f=%5CRightarrow%20%286x-1%29%3D%5Cpm%20%5Csqrt%7B289%7D)
![\Rightarrow (6x-1)=\pm 17](https://tex.z-dn.net/?f=%5CRightarrow%20%286x-1%29%3D%5Cpm%2017)
![\Rightarrow 6x=1\pm 17](https://tex.z-dn.net/?f=%5CRightarrow%206x%3D1%5Cpm%2017)
![\Rightarrow 6x=18 \:or\:6x=-16](https://tex.z-dn.net/?f=%5CRightarrow%206x%3D18%20%5C%3Aor%5C%3A6x%3D-16)
![\Rightarrow x=3 \:or\:x=-\frac{2}{3}](https://tex.z-dn.net/?f=%5CRightarrow%20x%3D3%20%5C%3Aor%5C%3Ax%3D-%5Cfrac%7B2%7D%7B3%7D)
We discard the negative value.
The side length of this rectangle is
![\Rightarrow l=(6(3)-1)=17,w=6(3)-1=17](https://tex.z-dn.net/?f=%5CRightarrow%20l%3D%286%283%29-1%29%3D17%2Cw%3D6%283%29-1%3D17)
QUESTION 3b
The given expression for the area of the rectangle is
.
This is equal to the indicated area which is ![1225\;in^2](https://tex.z-dn.net/?f=1225%5C%3Bin%5E2)
This implies that
![25x^2-50x+25=1225](https://tex.z-dn.net/?f=25x%5E2-50x%2B25%3D1225)
![25(x^2-2x+1)=1225](https://tex.z-dn.net/?f=25%28x%5E2-2x%2B1%29%3D1225)
![(5(x-1))^2=35^2](https://tex.z-dn.net/?f=%285%28x-1%29%29%5E2%3D35%5E2)
![5(x-1)5(x-1)=49](https://tex.z-dn.net/?f=5%28x-1%295%28x-1%29%3D49)
![\Rightarrow l=5(x-1),w=5(x-1)](https://tex.z-dn.net/?f=%5CRightarrow%20l%3D5%28x-1%29%2Cw%3D5%28x-1%29)
This implies that, the dimensions of the rectangle are equal;
Using the square root method, we obtain
![\Rightarrow 5(x-1)=\pm \sqrt{1225}](https://tex.z-dn.net/?f=%5CRightarrow%205%28x-1%29%3D%5Cpm%20%5Csqrt%7B1225%7D)
![\Rightarrow 5(x-1)=\pm 35](https://tex.z-dn.net/?f=%5CRightarrow%205%28x-1%29%3D%5Cpm%2035)
![\Rightarrow x=1\pm 7](https://tex.z-dn.net/?f=%5CRightarrow%20x%3D1%5Cpm%207)
![\Rightarrow x=8 \:or\:x=-6](https://tex.z-dn.net/?f=%5CRightarrow%20x%3D8%20%5C%3Aor%5C%3Ax%3D-6)
We discard the negative value.
The side length of this rectangle is
![\Rightarrow l=5(8-1)=35,w=5(8-1)=35](https://tex.z-dn.net/?f=%5CRightarrow%20l%3D5%288-1%29%3D35%2Cw%3D5%288-1%29%3D35)
QUESTION 3c
The given expression for the area of the rectangle is
.
This is equal to the indicated area which is ![289\;in^2](https://tex.z-dn.net/?f=289%5C%3Bin%5E2)
This implies that
![49x^2-56x+16=289](https://tex.z-dn.net/?f=49x%5E2-56x%2B16%3D289)
![(7x-4)^2=289](https://tex.z-dn.net/?f=%287x-4%29%5E2%3D289)
![(7x-4)(7x-4)=289](https://tex.z-dn.net/?f=%287x-4%29%287x-4%29%3D289)
![\Rightarrow l=(7x-4),w=(7x-4)](https://tex.z-dn.net/?f=%5CRightarrow%20l%3D%287x-4%29%2Cw%3D%287x-4%29)
Applying the laws of indices gives;
![(7x-4)^2=17^2](https://tex.z-dn.net/?f=%287x-4%29%5E2%3D17%5E2)
This implies that;
![7x-4=17](https://tex.z-dn.net/?f=7x-4%3D17)
![7x=21](https://tex.z-dn.net/?f=7x%3D21)
![x=3in.](https://tex.z-dn.net/?f=x%3D3in.)
The side length of this rectangle is
![\Rightarrow l=7(3)-4=17\:in.,w=7(3)-4=17\:in.](https://tex.z-dn.net/?f=%5CRightarrow%20l%3D7%283%29-4%3D17%5C%3Ain.%2Cw%3D7%283%29-4%3D17%5C%3Ain.)
Dont forget that the square is also a rectangle.