if alpha and beta are the zeros of the polynomial f(x)=ax2+bx+c then evaluate root b^2-4ac/c
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Answer:
If α and β are roots / zero of the quadratic equation
f(x)=ax
2
+bx
2
+c
aα+b
β
+
aβ+b
α
=
(aα+b)(aβ+b)
β(aβ+b)+α(aα+b)
=
a
2
αβ+abαabβ+b
2
aβ
2
+bβ+aα
2
+bα
=
a
2
(αβ)+ab(α+β)+b
2
a(α
2
+β
2
)+b(α+β)
observe, we have
α+β=
a
−b
αβ=
a
c
⇒ α
2
+β
2
=(α+β)
2
−2αβ
=
a
2
b
2
−
a
2c
Using equations on the right
=
a
2
(
a
c
)+ab(
a
−b
)+b
2
a(
a
2
b
2
−
a
2c
)+b(
a
−b
)
=
ac−b
2
+b
2
a
b
2
−2c−
a
b
2
=
ac
−2c
=
a
−2
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