If alpha,beta are the roots of x^2-12x+7=0, find the values of: alpha^2+beta^2

Question

zepdrix · Answer

Hmm this does NOT factor, so let's try throwing it into the Quadratic Formula:
$$\Large x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

dape · Answer

Well, if we already know the roots and call them $\alpha$ and $\beta$. We know they solve the equation, so we have
$$\alpha^2=12\alpha-7 \
\beta^2=12\beta-7$$
Adding these two, we get
$$ \alpha^2+\beta^2=12(\alpha+\beta)-14$$

tkhunny · Answer

I suppose, but this might be more interesting.

$\alpha\beta = 7$

$\alpha + \beta = 12$

Squaring the second: $\alpha^{2} + 2\alpha\beta + \beta^2 = 144$

Substitute from the first.  Are you seeing it, yet?

Interestingly, if you use the quadratic formula, square the two results, and add them, it results in 

$\alpha^{2} + \beta^{2} = \dfrac{b^{2} - 2ac}{a^{2}} = ??$

Unique answers don't care how you find them!

tkhunny · Answer

Note: More interesting than Brute Force!  About as interesting as @dape's solution.

dape · Answer

Right, if we use what @tkhunny said, namely $\alpha+\beta=12$, we can put it into my formula and get $\alpha^2+\beta^2=130$

dape · Answer

But there are numerous ways of doing it as you can see when you take everything into account.

anonymous · Answer

yeah i see. thankyou everyone