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1 answer:
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f(x) = 2 -4x
Step-by-step explanation:
Step 1 :
Given, f(x) = a(x - h)2 + k
Point on the parabola is (3, 6)
Vertex (h,k) = (1,-2)
Step 2:
Substituting the vertex in the equation we have,
f(x) = a(x-1)2 -2
Substituting the point (3,6) in this we have,
6 = a(3-1)2 - 2 => 6 = 4a -2
=> 4a = 8 => a = 2
Step 3 :
Substituting the value for a and the vertex in the given equation we have
f(x) = 2(x-1)2 -2 = 2(x2 - 2x + 1) -2 = 2x2 - 4x
=> f(x) = 2 -4x which is the standard form
Find the powers
$a^{2}=5+2 \sqrt{6}$
$a^{3}=11 \sqrt{2}+9 \sqrt{3}$
The cubic term gives us a clue, we can use a linear combination to eliminate the root 3 term $a^{3}-9 a=2 \sqrt{2}$ Square $\left(a^{3}-9 a\right)^{2}=8$ which gives one solution. Expand we have $a^{6}-18 a^{4}-81 a^{2}=8$ Hence the polynomial $x^{6}-18 x^{4}-81 x^{2}-8$ will have a as a solution.
Note this is not the simplest solution as $x^{6}-18 x^{4}-81 x^{2}-8=\left(x^{2}-8\right)\left(x^{4}-10 x^{2}+1\right)$
so fits with the other answers.