Find the Value of n/p for Quadratic Equations

In summary, the quadratic equation x^2 + mx + n = 0 has roots that are twice those of x^2 + px + m = 0 and the ratio of the roots is given by n/p = 8. This can be found by setting the ratio of the roots to 2 and using properties of the sum and product of roots in quadratic equations. This was an AMC 12B problem and the person discussing the problem did not make the USAMO but did advance to the AIME. They received a score of 8/15 on the AIME.
  • #1
lvlastermind
101
0
The quadratic equation x^2 + mx + n = 0 has roots that are twice those of x^2 + px + m = 0, and none of m, n, p is zero. What is the value of n/p?



I'm stuck and don't know where to begin.
 
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  • #2
Set the ratio of the roots to 2 and see whether u can isolate the ratio n/p

Daniel.
 
  • #3
In a quadratic equation [tex]ax^2 + bx + c = 0[/tex], there are two roots (either both real or both complex). The sum of the two roots is [itex]-\frac{b}{a}[/itex] and the product of the two roots is given by [itex]\frac{c}{a}[/itex].

Use this to come up with four simultaneous equations for the 2 pairs of roots for the given equations. The sum of the roots of the first equation is twice that of the sum of the roots of the second equation. The product of the roots of the first equation is 4 times the product of the roots of the second equation. Do a few algebraic manipulations to get the required ratio. Specifically, the property for the product of the roots will allow you to express n in terms of m, and the property for the sum of the roots will allow you to express m in terms of p, the rest is easy.
 
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  • #4
this was an amc 12B problem?
Anyway, the negative of roots add up to equal m, and mutliply to give n.
2(a+b)=m
(2a)(2b)=n

in the 2nd equation
(a)(b)=m
(a+b)=p

ab=2(a+b)...(a+b)=(ab)/2=p.
n=4ab
p=0.5ab
n/p=8
 
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  • #5
Yeah this was an amc question and I couldn't do it. Did you take the test? Did you advance to the next round?


p.s. Thanks for the help everyone
 
  • #6
yep, i went to the AIME (got a score of a 8/15), i didnt make the USAMO though. Did you make the AIME?
 
  • #7
yeah, i got a score of 5/15.
 

1. What is the value of n/p in quadratic equations?

The value of n/p in quadratic equations refers to the ratio between the coefficient of the linear term (n) and the coefficient of the quadratic term (p). It is often used to simplify or solve quadratic equations.

2. How do you find the value of n/p in a quadratic equation?

To find the value of n/p, you can use the quadratic formula or factor the quadratic equation. Once you have simplified the equation, the coefficient of the linear term (n) will be divided by the coefficient of the quadratic term (p) to get the value of n/p.

3. Why is finding the value of n/p important in quadratic equations?

The value of n/p is important in quadratic equations because it helps to determine the nature of the roots of the equation. If n/p is a rational number, the equation will have two real and distinct roots. If n/p is an irrational number, the equation will have two complex conjugate roots.

4. Can n/p ever be equal to 0 in a quadratic equation?

No, n/p cannot be equal to 0 in a quadratic equation because the denominator (p) cannot be equal to 0. This is because dividing by 0 is undefined and does not produce a meaningful result.

5. Are there any other ways to represent the value of n/p in quadratic equations?

Yes, instead of using the ratio n/p, the value can also be represented as the coefficient of the linear term (n) divided by the coefficient of the quadratic term (p) with a fraction bar or a colon. For example, n/p can also be written as n:p or n/p.

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