Question

The zeros of a parabola are −2 and −8. The maximum value of the function is 18. The parabola is drawn with a solid line, and the inside of the parabola is shaded. What quadratic inequality is represented by this description?

1. The quadratic inequality is represented by this description is y > 9x²/8 + 45x/4 + 369/8

### How to find the quadratic inequality represented by the description?

Since the function is a parabola, we represent is as y = f(x) = ax² + bx + c
Now, given that the zeros of a parabola are −2 and −8, we have that
f(-2) = 0
a(-2)² + b(-2) + c = 0
4a – 2b + c = 0
4a – 2b = -c     (1)
Also, f(-8) = 0
a(-8)² + b(-8) + c = 0
64a – 8b + c = 0
64a – 8b = -c    (2)
Also, dy/dx = d(ax² + bx + c)/dx = 2ax + b
At maximum value, dy/dx = 0.
So, 2ax + b = 0
2ax = -b
x = -b/2a
Substituting this into y, we have
f(-b/2a) = a(-b/2a)² + b(-b/2a) + c
f(-b/2a) = ab²/4a² – b²/2a + c
f(-b/2a) = b²/4a – b²/2a + c
f(-b/2a) = – b²/4a + c
Since the maximum value of y is 18. So,
f(-b/2a) = – b²/4a + c = 18
– b²/4a + c = 18
c = 18 + b²/4a
Substituting c into equation (1) and (2), we have
So, in equation (1)
4a – 2b = -c     (1) × 16
4a – 2b = -(18 + b²/4a)     (1)
4a – 2b = -18 – b²/4a
16a² – 8ab = -72a  (4)
Also, in equation (2)
64a – 8b = -c    (2)       ×  1
64a – 8b = -(18 + b²/4a)    (2)
64a – 8b = -18 – b²/4a)
256a² – 32ab = -72a  (5)
Subtracting equations (4) and (5),we have
16a² – 8ab = -72a  (4)
256a² – 32ab = -72a  (5)
-240a² + 24ab = 0
-24a(10a – b) = 0
⇒ 24a = 0 or 10a – b = 0
⇒ a = 0 or 10a = b
⇒ a = 0 or b = 10a
Substituting b = 10a into equation (4), we have
16a² – 8ab = -72a  (4)
16a² – 8a(10a) = -72a  (4)
16a² – 80a² = -72a  (4)
-64a² = -72a
-64a² + 72a = 0
-8a(8a – 9) = 0
⇒ -8a = 0 or 8a – 9 = 0
⇒ a = 0 or 8a = 9
⇒ a = 0 or a = 9/8
Substituting a into b, we have
b = 10a
b = 10 × 9/8
b = 5 × 9/4
b = 45/4
Substituting a nand b into c, we have
c = 18 + b²/4a
c = 18 + (10a)²/4a
c = 18 + 100a²/4a
c = 18 + 25a
c = 18 + 25 × 9/8
c = 18 + 225/8
c = (144 + 225)/8
c = 369/8
So, susbtituting the values of a, b and c into y, we have
y = ax² + bx + c
y =  9x²/8 + 45x/4 + 369/8

### The region represented by the shaded region

Since the equation of the parabola is y =  9x²/8 + 45x/4 + 369/8 and the region shaded is the inside of the parabola, we use the inequality sign > since the region is greater than y.
Also, since there is a solid line bounding the region, the line is not included in the inequality. so, the greater than sign > is used.
So, the shaded region is y > 9x²/8 + 45x/4 + 369/8
So, the quadratic inequality is represented by this description is y >  9x²/8 + 45x/4 + 369/8