F(x) = 0 and F(-x) = 0
If f ( x ) = 0 is a quadratic function which has two distinct real roots, then does f ( − x ) also have two distinct real roots?
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2 solutions
A quadratic equation always has two roots, either both real or both complex. The roots may be equal, when the discriminant is zero, but then we say that the equation has repeated roots. It should be mentioned that the roots are real (according to your argument) or complex.
Thx have updated the problem
Basically, x ↦ − x reflects the graph of f over the y-axis, so it does not change the number of times it crosses the x-axis.
Other solution :
If a 1 and a 0 are the two distinct roots of f , x ∈ R is a root of f ∘ ( − I d ) iff − I d ( x ) = − x is a root of f (ie x = − a 1 or − a 2 ).
We can generalise: for any φ bijection of R , if a 1 , a 2 , ⋯ are the zeros of f , the zeros of f ∘ φ are φ − 1 ( a 1 ) , φ − 1 ( a 2 ) , ⋯ .
Let f(x) = ax^2 + bx + c = 0 the equation has two roots if the discriminant is greater than 0. As it is given that f(x) has two roots it means that b^2 - 4ac > 0 f(-x) =0 is nothing but a*x^2 - bx + c = 0 the discriminant is still the same as before and so f(-x) also has two roots.