Why am I solving for x^2? I actually don't know what the final exact answer is, because of that.
I completed the square to solve this.
So I'm left with x = 1.5 ± sqrt(5.25). Do I solve for the individual numbers and then square them, or do I square both sides straight from that step?
Let a be equal to 1.5 + sqrt(5.25), and b be equal to 1.5 - sqrt(5.25). Thus, x=a and x=b. So if I square them, then I get a*a and b*b. But if I do the method where I square both sides, I get this:
[x = 1.5 ± sqrt(5.25)]^2.
I can't distribute the exponent, so we expand it:
(x = 1.5 ± sqrt(5.25))(x = 1.5 ± sqrt(5.25))
Either one could be a or b, so my answers are a*a, b*b, or a*b. (b*a = a*b, the commutative property of multiplication)
I plugged the calculation into desmos, and a=3.79128784748 and b=-0.791287847478, approximately. If I square them, I get a^2=14.3738635424 and b^2=0.626136457566. And quite nicely, a*b = -3.
Now, admittedly, the square root a*b is not a solution to the equation (it's imaginary). However, we're looking at what x^2 is, which is what made this problem so weird in the first place. And technically -3 fits that criteria, as since x=a and x=b, x^2 can be equal to a*b.
I really don't care if you give me a nerd emoji or not, but I'd like it if someone could tell me if one way is more right than the other or something.
Man I really I hope I didn't mess up something stupid.
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