The equation y = x^2 + \frac{1}{2}ax + a represents

Sereinserenormg

Sereinserenormg

Answered question

2022-01-30

The equation y=x2+12ax+a represents a parabola for all real values of a

Answer & Explanation

trovabile4p

trovabile4p

Beginner2022-01-31Added 13 answers

Let a, bR and ab. Show that the equation
x2+12ax+a=x2+12bx+b
has the unique solution x=2.
Gordon Stephens

Gordon Stephens

Beginner2022-02-01Added 10 answers

Just a disclaimer this is not the most formal proof so take it with some skepticism
Considered rearranging the quadratic so that a is the subject. You get a=yx2x2+1 since a can take any real value then the only candidates for x and y must make the right hand side an indeterminate form, e.g. since x and y are both finite both neumerator and denominator must be zero. So we want x2+1=0 which gives x=2 and yx2=0 which gives y=4. All that's left is to substitute these values into the original equation to verify that this is indeed a shared point.

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