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Quadratic Function in 2 variables and Mathematical Induction

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.

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?

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