Maths help

Question

1.x1 and x2 are positive integers of a quadratic equation x^2+ax+b=0. If 12,x1,x2 are the first three rates of an arithmetic line and x1,x2,4 are the first three rates of a geometric lin,then the value of a+b=...

 

a.54

b.48

c.39

d.34

e.15

Answer 1

I'm going to assume a couple of things here, I think the original question might have been worded a little differently. I'm assuming that the first statement means that x₁ and x₂ are positive integer solutions of the quadratic x²+ax+b=0.  Next, I'm going to assume that when it says an "arithmetic line" and a "geometric line," it means an arithmetic sequence and a geometric sequence.

 

So, let's start with the quadratic.  If x₁ and x₂ are solutions, then we know that

 

x²+ax+b=(x-x₁)(x-x₂).

 

If we multiply out the right hand side, we get

 

x²+ax+b=x²-(x₁+x₂)x+x₁x₂.

 

So, a=-(x₁+x₂) and b=x₁x₂.  So now we know what the relationships between x₁ and x₂ and a and b have to be.

 

Next, if 12, x₁, x₂ is an arithmetic sequence, that means that there must be a constant difference from one term to the next.  That means that

 

x₁-12=x₂-x₁.

 

If we rearrange that equation, we get

 

2x₁=x₂+12.

 

Now, let's look at the geomtric sequence.  That means that there must be a constant multiple from one term to the next.  If our sequence is x₁, x₂, 4, then

 

4/x₂=x₂/x₁.

 

If we rearrange that equation, we get

 

.5(x₂)²=2x₁.

 

Take the left hand side of that last equation and substitute it into the equation we got from the arithmetic sequence, and our newest equation reads

 

.5(x₂)²=x₂+12.

 

This is a quadratic.  Rearrange it to read

 

(x₂)²-2x₂-24=0,

 

which can be factored and solved to get x₂=4 or x₂=9.  If we try using x₂=4, the geometric sequence would tell us that x₁ must also be 4, but then the arithmetic sequence would have to be 12, 4, 4, which is not arithmetic.  If we try x₂=9, the geometric sequence tells us that x₁ must be 6, and putting that into the arithmetic sequence gives us 12, 9, 6, which does have a constant difference so both sequences work if x₁=9 and x₂=6.

 

Then, going back to the original relationships we found for a and b, we plug in the values of x₁ and x₂ to get a=-15 and b=54.  Then, a+b=-15+54=39.

 

Let me know if I can clarify any of that for you.