9. If x, Br. are zubes of folynomial 423-28 ²47, x match the follaing a) + B+x a b) * B+Br + roc ( ) ) / c) apr c ) C) is - 4 2 6 (نور)
Answers
9. If x, Br. are zubes of folynomial 423-28 ²47, x match the follaing a) + B+x a b) * B+Br + roc ( ) ) / c) apr c ) C) is - 4 2 6 (نور)
Answer:
search-icon-header
Search for questions & chapters
search-icon-image
Question
Bookmark
If a>b>0 and a
3
+b
3
+27ab=729 then the quadratic equation ax
2
+bx−9=0 has roots α,β(α<β). Find the value of 4β−aα.
Medium
Solution
verified
Verified by Toppr
a>b>0
Simplify the given equation.
a
3
+b
3
+27ab=729
⇒a
3
+b
3
+(−9)
3
−3ab(−9)=0
Using:- a
3
+b
3
+c
3
−3abc=(a+b+c)(a
2
+b
2
+c
2
−ab−bc−ca)
⇒(a+b−9)(a
2
+b
2
−ab+9a+9b+81)=0
therefore
a+b−9=0
a+b=9
Let, f(x)=ax
2
+bx+c
ax
2
+bx−9=0 →α+β=
a
−b
⇒αβ=
a
−9
(α<β)
a>b>0
f(1)=a+b−9
Thus it is clear that 1 is the root of given quadratic equation.
either α=1 or β=1
if β=1
α=
a
−9
We need
4β−aα=4×1−a(−
a
9
)
=4+9
=13.