quadratic equations Flashcards
discriminant
D=b^2-4ac
roots are given by
x=(-b±√D)/2a
nature of roots
(i) has real and distinct roots if and only if D>0.
(ii) has real and equal roots if and only if D=0.
(iii) has complex roots with non-zero imaginary parts if and only if D<0.
if p+iq is one root another root is
p-iq
if p+√q is an irrational root another root is
p-√q
If a,b,c∈Q and D is a perfect square. then αx^2+bx+c=0 then its root is
rational roots
If α and β are the roots of quadratic equation αx^2+bx+c=0; a≠0, then sum of roots =
α+β=
=-b/a
If α and β are the roots of quadratic equation αx^2+bx+c=0; a≠0, then product of roots are
=αβ=
=c/a
If α. Band γ are the roots of cubic equation ax^2+bx^2+cx+d=0;a≠0, then α+β+γ=
=-b/a
βγ+γα+αβ
=c/a
αβγ
=-d/a
Common Roots (Conditions) Suppose that the quadratic equations are ax^2+bx+c=0 and ax^2+b’x+c’=0
(i) When one root is common, then the condition is (a’c-ac’)^2=(bc’-b’c)(ab’-a’b)
(ii) When both roots are common, then the condition is a/a’=b/b’=c/c’
formation of a quadratic equation
If the roots of a quadratic equation are α and β, then the equation will be of the form x^2-(α+β)x+αβ=0.
formation of cubic equation
If α, β and y are the roots of the cubic equation, then the equation will be form of
x^3-(α+β+γ)x^2+(αβ+βγ+γα)x-αβγ=0
(i) When a>0, then minimum value of αx^2+bx+c is
(-D)/4a or (4ac-b^2)/4a at x=(-b)/2a
