Something to ponder. (1 Viewer)

Carrotsticks · Dec 11, 2012

Since this section is a bit quiet at the moment, here's something to promote a bit of discussion.

$\\ $We know that the quadratic discriminant for the polynomial $ P(x)=ax^2+bx+c $ (assume that a,b and c are real) is given by $ b^2-4ac. \\\\ $So for a quadratic to have no real roots, we must have $ b^2 - 4ac < 0. \\\\ $ Alternatively, we could let the roots be $ \alpha $ and $ \beta $ and analyse $ \alpha ^2 + \beta ^2. $ If we see that $ \alpha ^2 + \beta^2 < 0 $ then that means that both roots are complex. However, $ \alpha ^2 + \beta ^2 = \frac{b^2 - 2ac}{a^2} $ and if this is less than zero, we acquire $ b^2 - 2ac < 0 $ instead of our discriminant $ b^2 - 4ac < 0. \\\\ $Discuss why this is the case.$$

seanieg89 · Dec 11, 2012

Carrotsticks · Dec 11, 2012

I actually intended for students to discuss this (concept of "sufficient + necessary condition" or "iff"), but thanks for answering anyway lol.

RealiseNothing · Dec 11, 2012

It doesn't work both ways.

If $\alpha^2 + \beta^2 < 0$ then both roots are complex.

But if both roots are complex, then $\alpha^2 + \beta^2 < 0$ is not necessarily true.

Trebla · Dec 11, 2012

Carrotsticks said:
I actually intended for students to discuss this (concept of "sufficient + necessary condition" or "iff"), but thanks for answering anyway lol.

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Something to ponder. (1 Viewer)

Carrotsticks

Retired

seanieg89

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Carrotsticks

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RealiseNothing

what is that?It is Cowpea

Trebla

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