write a quadratic equation with zeroes √2 and√3
Answers
Answer:
hey mate here is ur answer...
The quadratic equation whose roots are (2+√3) and (2-√3) is x²-4x+1=0
Step-by-step explanation:
Let the quadratic equation be
ax²+bx+c=0, a≠0 and
\begin{lgathered}it's \: zeroes\:be\: \alpha \\ and \: \beta .\end{lgathered}
it
′
szeroesbeα
andβ.
Here, \alpha =2+\sqrt{3},and\:\beta = 2-\sqrt{3}Here,α=2+
3
,andβ=2−
3
\begin{lgathered}Sum \:of \: the \: roots \\= \alpha+\beta\\=2+\sqrt{3}+2-\sqrt{3}\\=4\end{lgathered}
Sumoftheroots
=α+β
=2+
3
+2−
3
=4
\begin{lgathered}Product \:of \: the \: roots \\= \alpha \beta\\=(2+\sqrt{3})(2-\sqrt{3})\\=2^{2}-(\sqrt{3})^{2}\\=4-3\\=1\end{lgathered}
Productoftheroots
=αβ
=(2+
3
)(2−
3
)
=2
2
−(
3
)
2
=4−3
=1
Therefore,
The quadratic equation is ax²+bx+c =0 is
x^{2}-(\alpha+\beta)x+\alpha \beta =0x
2
−(α+β)x+αβ=0
\implies x^{2}-4x+1=0⟹x
2
−4x+1=0
Step-by-step explanation: