Suppose p(x)=ax^2+bx+c is a quadratic polynomial satisfying p(7)−p(3)=214. Determine the value of p(9)−p(1).
p(7) - p(3) = 214
49a + 7b + c - (9a + 3b + c) = 214
40a + 4b = 214
p(9) - p(1)
= 81a + 9b + c - (a+b+c0
= 80a + 8b
= 2(40a + 4b)
= 2(214)= 428
To determine the value of p(9)−p(1), we need to find the quadratic polynomial p(x) and substitute x=9 and x=1 into it.
Given that p(x)=ax^2+bx+c, we can find the values of a, b, and c by using the information provided in the question.
p(7) can be calculated by substituting x=7 into the quadratic equation:
p(7) = a(7)^2 + b(7) + c
= 49a + 7b + c
Similarly, p(3) can be calculated by substituting x=3 into the quadratic equation:
p(3) = a(3)^2 + b(3) + c
= 9a + 3b + c
We are given that p(7)−p(3)=214, so we can write the equation as:
49a + 7b + c - (9a + 3b + c) = 214
Simplifying the equation:
40a + 4b = 214
Now, we have one equation with two variables. To solve for a and b, we will need another equation.
Let's consider the value of p(9)−p(1). We can substitute x=9 and x=1 into the quadratic equation:
p(9) = a(9)^2 + b(9) + c
= 81a + 9b + c
p(1) = a(1)^2 + b(1) + c
= a + b + c
Therefore, p(9)−p(1) can be calculated as:
p(9)−p(1) = (81a + 9b + c) - (a + b + c)
= 80a + 8b
Now, we can substitute the values of a and b in terms of the equation we obtained earlier (40a + 4b = 214).
Solving for a in terms of b, we get:
a = (214 - 4b) / 40
Substituting the value of a in terms of b into the equation p(9)−p(1):
p(9)−p(1) = 80((214 - 4b) / 40) + 8b
Simplifying further:
p(9)−p(1) = 2(214 - 4b) + 8b
= 428 - 8b + 8b
= 428
Therefore, the value of p(9)−p(1) is 428.