Quadratic equations

1) A = area, x = width and  l = 120 m - 2x

A = x(120 -2x)
A = 120x – 2x^2
A = -2x^2 + 120x
A = -2(x^2 – 60x)
A = -2(x^2 –80x + (-60/2)^2) + 2(30)^2
A = -2(x^2 –40x + (-30)^2) + 2(30)^2
A = -2(x - 30)^2 + 1800

The width = 30 m
The length = 120m – 2(30) m= 60 m
Area of largest rectangular of pasture = 1800 m^2 = 1800 sq. m
maximum because a is negative (a = -2)

2) 
Let l = length and w = width A = area of rectangle
2l + 4w = 800m
l + 2w = 400
l = 400 – 2w
A = l(w) length times width
A = (400 – 2w)w
A = -2w^2 + 400 w
A = -2(w^2 – 200w)
Completing square,
A = -2(w^2 – 200w + (-200/2)^2) + 2(-200/2)^2)
A = -2(w^2 – 200w + (-100)^2)) + 20000
A = -2(w- 100)^2 + 20000
Width = 100 m
L = 400 – 2w = 400 -2(100) = 200 m
Area of largest rectangular field = 20000 m^2 = 20000 sq. m
(Maximum because a is negative, a = -2)

3) 
7 = a(-2)^2 +b(-2) + 7
7 = 4a -2b + 7
4a – 2b = 0 ---------------- (1)
13 = a(1)^2 +b(1) + 7
a + b = 6 ------------------- (2)
Multiply equation (2) by 2
2a + 2b = 12 --------------- (3)

Add (1) and (3) together
6a = 12
a =12/6 = 2
From equation (2)
a + b = 6
2 + b = 6
b = 6-2 = 4
a = 2, b = 4

y = 2x^2 + 4x + 7
y = 2(x^2 + 2x) + 7
Completing Square
y = 2(x^2 + 2x + (2/2)^2)) + 7 -2(1)^2
y = 2(x^2 + 2x + (1)^2) + 7 -2(1)^2
y = 2(x+1)^2 + 7 -2
y = 2(x+1)^2 + 5

The minimum value is 5 (it is a minimum because a is positive, a = +2)


4)
At point (-2, 0)
y = ax^2 + 2x + c
0 = a(-2)^2 + 2(-2) + c
0 = 4a -4 + c
4a + c = 4 --------------- (1)
At point (4, 0)
y = ax^2 + 2x + c
0 = a(4)^2 + 2(4) + c
0 = 16a + 8 + c
16a + c = -8 ------------- (2)
Equation (2) subtracts equation (1)
12a = -12
a = -12/12 = -1
From equation (1)
4a + c = 4
4(-1) + c = 4
c = 8
a = -1, c = 8

y = -(x^2 - 2x) + 8
y = -[x^2 - 2x + (-2/2)^2)] + 8 + (-2/2)^2
y = -[x^2 - 2x + (-1)^2)] + 8 + 1
y = -(x- 1]^2 + 9

The maximum value of y = 9. It is a maximum because a is negative. a = -1