Question

In the quadratic equations xa×xb=0 and xc×xd=0, c and d are the roots of the first equation, and a and b are the roots of the second equation, respectively. Find all possible non-zero real values of a,b,c,d.

a)a=b=c=d
b) a=−b=c=−d
c) a=−c, b=−d
d) a=−d, b=−c

Answer 1

Option (b) a=-b=c=-d is the correct relationship between the roots of the given quadratic equations, as it satisfies the conditions for sums and products of roots in each equation.

Step-by-step explanation:

We are given two quadratic equations where the roots of one are the coefficients of the other. The equations are x×x×b=0 and xc×xd=0, where c and d are the roots of the first equation, whereas a and b are the roots of the second equation. By the property of quadratic equations, if p and q are the roots of the quadratic equation ax²+bx+c=0, then p+q=-b/a and pq=c/a. Applying this to the equations given:

For x×x×b=0: a=-1, b (the product of roots) is unknown.

For xc×xd=0: c=-1, d (the product of roots) is unknown.

Consequently, the roots of the first equation (a and b) must add up to 1 and multiply to b, while the roots of the second equation (c and d) must add up to 1 and multiply to d. Therefore, option (b) a=−b=c=−d represents the correct relationship between the roots, as it satisfies the required conditions for the sums and products of the roots of the respective quadratic equations.