Explore BrainMass

Explore BrainMass

    Quadratic Function in 2 variables and Mathematical Induction

    Not what you're looking for? Search our solutions OR ask your own Custom question.

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    1) Prove that in the equation x^2 - 3xy +2y^2 - 2x - 3y - 35 = 0, for every real value of y there is a real value of x, and for every real value of x there is a real value of y.

    2) Use the Principle of Mathematical Induction to prove:
    a) For every positive integer n, 4^(2n + 1) + 3^(n+2) is a multiple of 13.
    b) 3^n > n^2 for n >= 1

    3) Prove that if 3 is a multiple of b^2 then 3 is also a multiple of b.

    © BrainMass Inc. brainmass.com December 15, 2022, 8:09 pm ad1c9bdddf
    https://brainmass.com/math/basic-algebra/quadratic-function-variables-mathematical-induction-276883

    Solution Preview

    1) Prove that in the equation x^2 - 3xy +2y^2 - 2x - 3y - 35 = 0, for every real value of y there is a real value of x, and for every real value of x there is a real value of y.

    quadratic equation is ax^2 + bx +c =0

    The equation has real solutions if the discriminant = b^2 - 4ac is >=0

    This is a quadratic equation in x and y.

    If the unknown is x, this can be written as

    x^2 + (-3y -2) x + 2y^2 - 3y - 35 = 0

    It has real solutions, if the discriminant is positive.

    That is if (3y + 2) ^2 - 4 x 1 x (2y^2 - 3y - 35) >= 0

    9y^2 + 4 + 12y - 8y^2 + 12y + 140

    y^2 + 24y + 144 which is (y+12)^2 which is a ...

    Solution Summary

    How to compute the domain and range of a quadratic function and find when it takes real values?

    Proofs for mathematical induction problems. How to solve any problem by the principle of mathematical induction?

    $2.49

    ADVERTISEMENT