- #1
zooxanthellae
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Quadratic "Proof"
Show that if [tex]a>0[/tex], then [tex]ax^2 + bx + c \geq 0[/tex] for all values of [tex]x[/tex] if and only if [tex]b^2 - 4ac \leq 0[/tex]
http://www.math.toronto.edu/~drorbn/classes/0405/157AnalysisI/HW2/HW.html
I believe I'm supposed to be working only with basic rules like the commutative and distributive laws.
I re-arranged [tex]b^2 - 4ac \leq 0[/tex] to [tex]b^2 \leq 4ac[/tex].
Then I took the square root of both sides to get [tex]b \leq 2\sqrt{ac}[/tex].
I thought about this result and concluded that this meant that [tex]b/2[/tex] is less than some number between [tex]a[/tex] and [tex]c[/tex]. But I couldn't see where to go from there.
Then I tried working with the equation I was given in order to see if I could manipulate it so that [tex]b^2 - 4ac \leq 0[/tex] would have to be true.
So I re-arranged [tex]ax^2 + bx + c \geq 0[/tex] to [tex]ax^2 + c \geq |-bx|[/tex]. And then I found myself stuck once again.
I think my problem is with the [tex]x^2[/tex] and [tex]x[/tex]. I don't see how the preconditions given really "address" the difference between the two.
Homework Statement
Show that if [tex]a>0[/tex], then [tex]ax^2 + bx + c \geq 0[/tex] for all values of [tex]x[/tex] if and only if [tex]b^2 - 4ac \leq 0[/tex]
http://www.math.toronto.edu/~drorbn/classes/0405/157AnalysisI/HW2/HW.html
Homework Equations
I believe I'm supposed to be working only with basic rules like the commutative and distributive laws.
The Attempt at a Solution
I re-arranged [tex]b^2 - 4ac \leq 0[/tex] to [tex]b^2 \leq 4ac[/tex].
Then I took the square root of both sides to get [tex]b \leq 2\sqrt{ac}[/tex].
I thought about this result and concluded that this meant that [tex]b/2[/tex] is less than some number between [tex]a[/tex] and [tex]c[/tex]. But I couldn't see where to go from there.
Then I tried working with the equation I was given in order to see if I could manipulate it so that [tex]b^2 - 4ac \leq 0[/tex] would have to be true.
So I re-arranged [tex]ax^2 + bx + c \geq 0[/tex] to [tex]ax^2 + c \geq |-bx|[/tex]. And then I found myself stuck once again.
I think my problem is with the [tex]x^2[/tex] and [tex]x[/tex]. I don't see how the preconditions given really "address" the difference between the two.