Quadratic Function in 2 variables and Mathematical Induction
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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.
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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?
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 ...
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