Which of the following is a quadratic equation whose roots are 2+√3 and 4-2√3 ?

Correct option is A

Given:

Roots: $$\alpha$$ = $$2+\sqrt{3}$$ and $$\beta$$ = $$4-2\sqrt{3}$$

​​Formula:

The quadratic equation with roots α and β is:

​$$x^2 - ( \alpha+\beta)x +\alpha\beta=0$$

Solution:

Sum of the roots ($\alpha +\beta$)

​$$α+β=(2+ \sqrt3​ )+(4−2 \sqrt3​ )$$

​​$$\alpha+\beta = 6-\sqrt{3}$$

​​Product of the roots ($\alpha \beta$):

​$$αβ=(2+ \sqrt3​ )(4−2 \sqrt3​ )$$

​​using distributive property

​$$αβ=(2⋅4)+(2⋅−2 \sqrt3​ )+( \sqrt3​ ⋅4)+( \sqrt3​ ⋅−2 \sqrt3​ )$$

​​$$αβ=8−4 \sqrt3​ +4 \sqrt3​ −6$$

​​$$αβ=8−6=2$$

​​Substitute ​​$$\alpha+\beta = 6-\sqrt{3}$$ and $$\alpha\beta=2$$

​$$x^2 - ( \alpha+\beta)x +\alpha\beta=0$$​​

​$$x ^2 −(6− \sqrt3​ )x+2=0$$

Thus, correct answer is (a) $$x ^2 −(6− \sqrt3​ )x+2=0$$​​