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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Kennedy 6 days 2021-10-08T06:29:52+00:00 1 Answer 0

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    2021-10-08T06:30:57+00:00

    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.

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