Expert: Ahmed Salami - 9/17/2011

Question
Find the quadratic equation x^2 + px + q = 0 if the roots a and b are the dimensions of a rectangle whose diagonal is 8 and area is 24.

Answer
Hi Chris,
The area of the rectangle,
A = ab
24 = ab
The diagonal D is related to a and b by the Pythagoras theorem;
a² + b² = D²
a² + b² = 8² = 64
Now, if a and b are the roots of a quadratic equation
(x - a)(x - b) = 0
x² - (a+b)x + ab = 0
So,
q = ab = 24
p = -(a+b)
squaring
p² = (a+b)²
= (a² + b²) + 2ab
= 64 + 2(24)
= 112
p = √112
= -4√7
Note that we take p as the negative value because a and b are positive.
The equation is therefore,
x² - (4√7)x + 24 = 0

Regards