Answer:
The other solution of the given equation x² + bx + c = 0 is also rational number.
Step-by-step explanation:
Here, given: ax² + bx + c = 0 is a quadratic equation
Also, one solution (r) of equation is RATIONAL.
To show: The other solution (s) is also RATIONAL
Now, here: as x² + bx + c = 0
Since r and s are the two given solutions, the given equation can be factorized as:
x² + bx + c = (x -r) (x - s)
Simplifying LHS, we get:
(x -r) (x - s) = x x - r (x) - s (x) + (r)(s)
= x² + x(-r - s) + rs
or, x² + bx + c = x² + x(-r - s) + rs
Comparing the related terms, we get:
b = (-r - s)
⇒ b + s = - r
or, s = -r - b
Now, given : r = Rational and the negative of a rational is also rational.
⇒ -r is also rational
Also, difference of two rational number is also rational.
⇒ -r - b is also rational
⇒ s is a RATIONAL NUMBER
Hence, the other solution of the given equation x² + bx + c = 0 is also rational number.
