Find the Number of Solutions to Simple Equations

Algebra Level 3

$\left\{\begin{array}{cc}a+b+c+d &=& 20,\\ ab+ac+ad+bc+bd+cd &=& 150.\end{array}\right.$

Find the number of positive ordered quadruplets $(a,b,c,d)$ that satisfy the two equation above.

1 solution

Alan Yan
Oct 14, 2015

It is easy to see that $a^2 + b^2 + c^2 + d^2= 100$

By $QM - AM$ , we have that $\sqrt{\frac{a^2 + b^2 + c^2 + d^2}{4}} \geq \frac{a+b+c+d}{4}$

However they are equal. Equality holds when $a = b = c = d = 5$ and this is the only solution.

Exactly Same Way

Kushagra Sahni - 5 years, 8 months ago

what is qm please do explain

Kaustubh Miglani - 5 years, 8 months ago

Quadratic Mean.

Kushagra Sahni - 5 years, 8 months ago

so what is the formula for this

Kaustubh Miglani - 5 years, 8 months ago

@Kaustubh Miglani – Quadratic mean is what Alan has written on the RHS

Kushagra Sahni - 5 years, 8 months ago

@Kushagra Sahni – is it the sum of squares of observations divided by no. of observations

Kaustubh Miglani - 5 years, 7 months ago

@Kaustubh Miglani – $(a+b+c+d)^2 = a^2 + b^2 + c^2 + d^2 + 2(ab + ac + ad + bc + bd + cd)$

Alan Yan - 5 years, 7 months ago