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kifflom [539]
3 years ago
8

Determine the largest integer value of a which f(x)= ax^2 + 9x + 5 has two distinct, real zeros (show work please)

Mathematics
1 answer:
levacccp [35]3 years ago
7 0
First, understand the question. We want to find the biggest integer (whole number) for a so that f(x) = ax^2 + 9x + 5 has two distinct, real zeros (it crosses the x-axis twice).

Several ways to approach this one.
1) I would begin by graphing the equation on Desmos.com. I tried several values for a (looking for the biggest one) and I found that the biggest integer value appears to be around 4.

2) Now, we need to prove that this is true by showing our work. But how? 

We first, ask:
Q: "How can I find the roots or zeros of a quadratic equation?"
A: One way is by using the quadratic formula!

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

For our equation, the quadratic formula would be:
x = \frac{-9 \pm \sqrt{9^2 - 4(a)(5)}}{2(a)}

Now, remember that in order to have two real roots, the "discriminant" (the inside of the square root) MUST be positive. So what we're saying is that:
81 - 20 a > 0

But that's the same as:
81 > 20a

Or
4.05 > a

So, we are saying that in order to have two real roots, a must be less than 4.05. The biggest integer we can pick that is less than 4.05 is 4 !!! We are done! We have proven that a = 4.
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Answer:

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Step-by-step explanation:

In short, the sum of the opposite areas are equal.

x + 30 = 24 + 48

x = 42

To prove this, draw a line from each corner to the "center" where the four lines meet.  Along each side of the square are two triangles.  These triangles have the same base and the same height, and therefore have the same area.

If we say the triangles at the bottom have area a, the triangles on the left have area b, the triangles on top have area c, and the triangles on the right have area d, then we can write 4 equations:

a + b = x

b + c = 24

c + d = 30

a + d = 48

Adding the first and third equations:

a + b + c + d = x + 30

Adding the second and fourth equations:

a + b + c + d = 24 + 48

Therefore:

x + 30 = 24 + 48

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