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?

Answers

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

quadratic inequalityis represented by this description is y > 9x²/8 + 45x/4 + 369/8## How to find the quadratic inequality represented by the description?

parabola, we represent is as y = f(x) = ax² + bx + cparabolaare −2 and −8, we have that## The region represented by the shaded region

quadratic inequalityis represented by this description is y > 9x²/8 + 45x/4 + 369/8quadratic inequalityhere: