what is the factor of a^4-7a^2+9 ?
a^4-7a^2+9
a^2 = (7±√13)/2
Now it gets tricky:
7+√13 = (1+2√13+13)/2 = (1+√13)^2/2
So, if
a^2 = (7+√13)/2 = (1+√13)^2/4
a = ±1/2 (√13+1)
and
a = ±1/2 (√13-1)
Makes things kind of nasty, the factors are
1/16 (±2x + √13 ± 1)
where all 4 combinations of + and - are used.
Hint:
substitute u=a^2, then
a^4-7a^2+9
=u^2-7u+9
and try to factorize.
However, check you question to see if there is a typo. The above expression in u does not have rational factors.
To find the factors of the given expression a^4 - 7a^2 + 9, we can use factoring by grouping. The expression has three terms, so we'll group the terms in pairs and factor out common factors from each pair.
Step 1: Group the terms
a^4 - 7a^2 + 9 = (a^4 - 4a^2) + (-3a^2 + 9)
Step 2: Factor out common factors from each pair
(a^4 - 4a^2) + (-3a^2 + 9) = a^2(a^2 - 4) - 3(a^2 - 3)
Now, we have (a^2 - 4) and (a^2 - 3) as common factors. We can further factor these terms.
Step 3: Factor the common factors
a^2(a^2 - 4) - 3(a^2 - 3) = a^2(a + 2)(a - 2) - 3(a - √3)(a + √3)
The factored form of the expression a^4 - 7a^2 + 9 is: (a^2 + 2)(a^2 - 2) - 3(a - √3)(a + √3)