Q. If alpha and beta are roots of equation 3x^2+2x-5=0, how would you form a quadratic equation whose roots are α/β^2 and β/α ^2?
Solution:-
Given that the Quadratic equation is 3x^2+2x-5=0
and root is α and β
α + β =-b/a=-2/3
α β =c/a=-5/3
Formation of Quadratic equation by the root α/β^2 and β/α ^2
sum of root
α/β^2 + β/α^2
=α^3+β^3/α^2β^2
=(α + β) (α^2 - αβ+β^2) /α^2β^2
=(α + β) (α^2- αβ+β^2) /α^2β^2
=(α + β) {(α + β)^2- 2αβ-αβ}/α^2β^2
=(α + β) {(α + β)^2- 3αβ}/α^2β^2
=(-2/3) {(-2/3)^2- 3(-5/3)}/(-5/3)^2
=(-2/3) (4/9 +5)/25/9
= (-2/3)(49/9)/25/9
=-98/15
Product of the root
=(α/β^2) (β/α^2)
=1/αβ
=1/-5/3
=-3/5
=multiplying in numerator and denumerator by 3
=-9/15
so, a=15 b=98 , c=-9
new quadratic Equation is
ax^2+bx +c=0
15x^2+98x+(-9)=0
15x^2+98x-9=0
Q.If α and β are the roots of the equation 2x2−3x−6=0, then the equation whose roots are α^2+2,β^2+2, is
Solution:-
Given that the Quadratic equation is 2x^2−3x−6=0
and root is α and β
α + β =-b/a=-(-3/2)=-3/2
α β =c/a=-6/2
Formation of Quadratic equation by the root α^2+2 and β ^2+2
sum of root
α^2+2 + β ^2+2 =α^2+ β ^2 +4
(α + β)^2- 3αβ
Next comming soon
Q.Check whether the given equation is quadratic equation or not?
7x=2x×x
Solution
Given that the equation
⇒ 7x=2x×x
⇒ 7x=2x^2
⇒ 2x^2 - 7x=0 yes this Quadratic Equation because Maximum power of x is 2.
