SOLUTION: Given that {{{a}}} and {{{ma}}} are the roots of the equation {{{x^2+px+q=0}}}, show that {{{mp^2 = (m+1)^2q}}}.

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equation Customizable Word Problems -> SOLUTION: Given that {{{a}}} and {{{ma}}} are the roots of the equation {{{x^2+px+q=0}}}, show that {{{mp^2 = (m+1)^2q}}}.      Log On


   

Question 1146314: Given that a and ma are the roots of the equation x%5E2%2Bpx%2Bq=0, show that mp%5E2+=+%28m%2B1%29%5E2q.
Found 2 solutions by KMST, MathTherapy:
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
If thr roots of a quadratic equation are a and ma ,
the equation can be written in factored form as
%28x-a%29%28x-ma%29=0 .
Doing the multiplication we find that
x%5E2-%28ma%2Ba%29x%2Ba%28ma%29=0 or x%5E2-%28m%2B1%29ax%2Bma%5E2=0
is the equivalent form of the quadratic equation,
in the form x%5E2%2Bpx%2Bq=0 , with p=-a%28m%2B1%29 and q=ma%5E2 .
Then,
mp%5E2=m%28a%28m%2B1%29%29%5E2 , which simplifies to mp%5E2=ma%5E2%28m%2B1%29%5E2 ,
which is obviously %28m%2B1%29%5E2q=%28m%2B1%29%5E2%28ma%5E2%29 .

Answer by MathTherapy(10549) About Me  (Show Source):
You can put this solution on YOUR website!

Given that a and ma are the roots of the equation x%5E2%2Bpx%2Bq=0, show that mp%5E2+=+%28m%2B1%29%5E2q.
Since “a” and “ma” are roots of matrix%281%2C3%2C+x%5E2+%2B+px+%2B+q%2C+%22=%22%2C+0%29, then we can say that:

      Sum of roots = a + ma

Also, sum of roots also = matrix%281%2C5%2C+%28-+b%29%2Fa%2C+%22=%22%2C+%28-+p%29%2F1%2C+%22=%22%2C+-+p%29

Therefore, a + ma = - p 

a(1 + m) = - p

matrix%281%2C3%2C+a%2C+%22=%22%2C+%28-+p%29%2F%281+%2B+m%29%29 ------ eq (i)



Product of roots = matrix%281%2C3%2C+a%28ma%29%2C+or%2C+a%5E2m%29

Product of roots also = matrix%281%2C5%2C+c%2Fa%2C+%22=%22%2C+q%2F1%2C+%22=%22%2C+q%29

Therefore, matrix%281%2C3%2C+a%5E2m%2C+%22=%22%2C+q+%29

matrix%281%2C3%2C+a%5E2%2C+%22=%22%2C+q%2Fm%29 ------ eq (ii)

matrix%281%2C3%2C+%28%28-+p%29%2F%281+%2B+m%29%29%5E2%2C+%22=%22%2C+q%2Fm%29 -------- Substituting %28++-++p%29%2F%281+%2B+m%29 for a in eq (ii)

matrix%281%2C3%2C+p%5E2%2F%281+%2B+m%29%5E2%2C+%22=%22%2C+q%2Fm%29

highlight_green%28matrix%281%2C3%2C+mp%5E2%2C+%22=%22%2C+%281+%2B+m%29%5E2q%29%29 ------- Cross-multiplying