if a and b are the roots of equation x²-4x+1=0 find 1} (a²+b²) .2} a³+b³ 3} a³-b³ 4} a/b + b/a
Answers
Step-by-step explanation:
Step-by-step explanation:
GIVEN:
\alphaα &\betaβ are the roots of x²–4x+3.
TO FIND:
\alpha^{2}α
2
+\beta^{2}β
2
and a³+B³
ANSWER:
The given equation is a Quadratic equation.
So, we can find it's roots by either Factorising or by standard formula.
On Factorising:
x^{2}-4x+3x
2
−4x+3
=x^{2}-x-3x+3x
2
−x−3x+3
=x(x-1) -3(x-1)x(x−1)−3(x−1)
=(x-3) (x-1)(x−3)(x−1)
On equating with 0.
(x-3) (x-1) =0(x−3)(x−1)=0
Therefore x = 3,1.
\large\red{\boxed{ x =2, 1}}
x=2,1
\alpha=3α=3
\beta=1β=1
So, \alpha^{2}α
2
= 3×3=9
\beta^{2}β
2
=1×1=1
Hence, \large\purple{\boxed{\alpha^{2}+ \beta^{2} =9+1=10}}
α
2
+β
2
=9+1=10
And,
\alpha^{3}α
3
=3×3×3=27
\alpha^{3}α
3
=3×3×3=27
\beta^{3}β
3
=1×1×1=1
Hence
\large\green{\boxed{\alpha^{3}+\beta^{3}=27+1=28.}}
α
3
+β
3
=27+1=28.
We can also find roots by using standard formula
that is
\large\blue{\boxed{\dfrac{-b+-\sqrt{b^{2}-4ac}}{2a}}}
2a
−b+−
b
2
−4ac