if alpha and bita are the zeroes of ax^2+bx+c then evaluate alpha^4+bita^4
I will use m and n for alpha and beta
from the equation we know
m+n = -b/a
mn = c/a
(m+n)^2 = m^2 + n^2 + 2mn
so m^2 + n^2 = (m+n)^2 - 2mn
also
(m^2 + n^2)^2 = m^4 + n^4 + 2m^2n^2
so m^4 + n^4 = (m^2 + n^2)^2 - 2(mn)^2
=[(m+n)^2 - 2mn]^2 - 2(mn)^2
= [(-b/a)^2 - 2c/a]^2 - 2c^2/a^2
= [b^2/a^2 - 2c/a]^2 - 2c^2/a^2
let's test it with x^2 - 7x + 12 = 0
roots m and n , where m = 4, n=3
m+n= 7
mn = 12
according to my answer
m^4 + n^4 = [(49/1-2(12)]^2 - 2(144))/1 = 25^2 - 288 = 337
actual m^4 +n^4 = 4^4 + 3^4 = 256 + 81 = 337
YEAH!
For ax^2+bx+c=0
the zeroes are:
α,β = (-b±sqrt(b²-4ac))/2a
=(p±q)
where
p=-b/2a
q=sqrt(b²-4ac)/2a
So to evaluate
α^4+β^4
=(p+q)^4+(p-q)^4
=2(p^4+6p²q²+q^4)
=2[(-b/2a)^4+6(-b/2a)²(sqrt(b²-4ac)/2a)²+(sqrt(b²-4ac)/2a)^4]
Expand and simplify to get:
2c²/a²-4b²c/a³+(b/a)^4
Check me.
If α and β are zero of f(x)=ax
2
+bx+c. The evaluate α
4
×β
4
Note that we have
=(α+β)
4
=α
4
+β
4
+
4
C
1
α
3
β
1
+
4
C
2
α
2
β
2
+
4
C
3
α
1
β
1
3
Then, α
4
+β
4
=(α+β)
4
−4(α
3
β
1
+α
1
β
3
)−6α
2
β
2
=(α+β)
4
−4αβ(α
2
+β
2
)−6(αβ)
2
Maxover, we have
α+β=−
a
b
and αβ=
a
c
Then,
α
4
+β
4
=(−
a
b
)
4
−4(
a
c
)[(−
a
b
)
2
−2(
a
c
)]−6(
a
c
)
2
=
a
4
b
4
−
a
4c
(
a
2
b
2
)+8(
a
c
)
2
−6(
a
c
)
2
=
a
4
b
4
−
a
3
4b
2
c
+2
a
2
C
2
To evaluate α^4 + β^4, we first need to find the values of α and β. Given that α and β are the zeroes of the quadratic equation ax^2 + bx + c, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For the equation ax^2 + bx + c = 0, the sum of the zeroes (α + β) is equal to -b/a, while the product of the zeroes (α * β) is equal to c/a.
Let's use this information to find the value of α^4 + β^4:
α^4 + β^4 = (α^2)^2 + (β^2)^2
We can express α^2 and β^2 in terms of α + β and α * β using the identity:
(α + β)^2 = α^2 + 2αβ + β^2
(α - β)^2 = α^2 - 2αβ + β^2
Expanding (α + β)^2 and rearranging the terms, we have:
α^2 + 2αβ + β^2 = (α + β)^2
Similarly, expanding (α - β)^2, we have:
α^2 - 2αβ + β^2 = (α - β)^2
From the quadratic equation, we know that α + β = -b/a and α * β = c/a. Substituting these values, we get:
α^2 + 2αβ + β^2 = (-b/a)^2
α^2 - 2αβ + β^2 = (2c/a)^2
Now, we can substitute these values back into the expression of α^4 + β^4:
α^4 + β^4 = (α^2)^2 + (β^2)^2
= (α^2 + 2αβ + β^2)(α^2 - 2αβ + β^2)
= (-b/a)^2 * (2c/a)^2
= b^2 * (4c^2/a^2)
= 4b^2c^2 / a^2
Therefore, α^4 + β^4 = 4b^2c^2 / a^2.