## Prove that if one solution for a quadratic equation of theform x2 + bx + c = 0 is rational (where b and c are rational),then the other solut

Question

Prove that if one solution for a quadratic equation of theform x2 + bx + c = 0 is rational (where b and c are rational),then the other solution is also rational. (Use the factthat if the solutions of the equation are r and s, thenx2 + bx + c = (x − r )(x − s).)

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2021-10-08T06:29:52+00:00
2021-10-08T06:29:52+00:00 1 Answer
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## Answers ( )

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 RATIONALNow, 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) + rsor,

x² + bx + c = x² + x(-r – s) + rsComparing 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 NUMBERHence, the other solution of the given equation x² + bx + c = 0 is also rational number.