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FrogPad
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Umm... I guess I don't remember this from algebra, but I have a rather basic question.
Let's say you have the expression,
[tex] f(x) = 4x^2 + 5x + 1 [/tex]
And we want to find the roots.
So,
[tex] \frac{-5 \pm \sqrt{5-4(4)}}{2(4)}= \{ \frac{1}{2},-1 \} [/tex]
Now, if we plug these into [itex] f(x) [/itex] we have:
[tex] f(1/2) = 9/2 [/tex]
[tex] f(-1) = 0 [/tex]
So obviously the quadratic equation does not yield the roots. But if we let [itex] a=1 [/itex].
[tex] 4(x^2+\frac{5}{4}x+\frac{1}{4}) [/tex]
Again, using the quadratic:
[tex] x = \{ -1,-\frac{1}{4} \} [/tex]
So now plugging this into [itex] f(x) [/itex]
[tex] f(x) = 4(x^2+\frac{5}{4}x+\frac{1}{4}) [/tex]
yields:
[tex] f(-1) = 0 [/tex]
[tex] f(\frac{-1}{4}) = 0 [/tex]
So what gives. I don't think I was ever taught that "a" has to be equal to 1 for it to hold.
If someone could clarify why this is that would be awesome. I seem to remember something about completing the square, but the thing is, is I have an engineering test on Thursday, and the last thing I want to be using my time on is figuring out why this is. But I'm curious
Thanks
Let's say you have the expression,
[tex] f(x) = 4x^2 + 5x + 1 [/tex]
And we want to find the roots.
So,
[tex] \frac{-5 \pm \sqrt{5-4(4)}}{2(4)}= \{ \frac{1}{2},-1 \} [/tex]
Now, if we plug these into [itex] f(x) [/itex] we have:
[tex] f(1/2) = 9/2 [/tex]
[tex] f(-1) = 0 [/tex]
So obviously the quadratic equation does not yield the roots. But if we let [itex] a=1 [/itex].
[tex] 4(x^2+\frac{5}{4}x+\frac{1}{4}) [/tex]
Again, using the quadratic:
[tex] x = \{ -1,-\frac{1}{4} \} [/tex]
So now plugging this into [itex] f(x) [/itex]
[tex] f(x) = 4(x^2+\frac{5}{4}x+\frac{1}{4}) [/tex]
yields:
[tex] f(-1) = 0 [/tex]
[tex] f(\frac{-1}{4}) = 0 [/tex]
So what gives. I don't think I was ever taught that "a" has to be equal to 1 for it to hold.
If someone could clarify why this is that would be awesome. I seem to remember something about completing the square, but the thing is, is I have an engineering test on Thursday, and the last thing I want to be using my time on is figuring out why this is. But I'm curious
Thanks
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