Let for a≠a1≠0 f(x) = ax^2 + bx + c, g(x)= a1x^2 + b1X + C1and p(x) = f(x) – g(x). If p(x) = 0 only for x=-1 and p(-2) = 2, then the value of p(2) is
(1)3
(2)9
(3) 6
(4) 18
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Answer:
Correct option is
B
18
Given, f(x)=ax
2
+bx+c,g(x)=ax
2
+bx+c,
and p(x)=f(x)−g(x)
p(x)=0 for only x=−1,p(−2)=2 and a
=a
1
=0
∴p(x)=(a−a
1
)x
2
+(b−b
1
)x+(c−c
1
)
The standard quadratic equation is ax
2
+bx+c=0
For roots to be equal ⇒b
2
−4ac=0
Then Sum of roots = −
a
b
Here, Only root is −1⇒D=0⇒(b−b
1
)
2
=4(a−a
1
)(c−c
1
)→(1)
Sum of roots ⇒(−1)+(−1)=
(a−a
1
)
−(b−b
1
)
⇒(b−b
1
)=2(a−a
1
)→(2)
From (1) and (2), c−c
1
=(a−a
1
)→(3)
⇒p(x)=(a−a
1
)x
2
+2(a−a
1
)x+(a−a
1
)
Now, p(−2)=2
⇒2=(a−a
1
)(−2)
2
−2(a−a
1
)(−2)+(a−a
1
)
⇒(a−a
1
)=2
Now p(2)=2(2)
2
+2(2)(2)+2=18
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