Q. 104.6( 14 Votes )
If f(x) = ax2 + bx + c and f(0) = 2, f(1) = 1 and f(4) = 6, then let us calculate the values of a, b and c.
Answer :
We have,
f(x) = ax2 + bx + c …(i)
f(0) = 2
f(1) = 1
f(4) = 6
Replace x by 0 in equation (i), we get
f(0) = a(0)2 + b(0) + c
⇒ 2 = 0 + 0 + c [∵ f(0) = 2]
⇒ 2 = c
⇒ c = 2 …(ii)
Now, replace x by 1 in equation (i), we get
f(1) = a(1)2 + b(1) + c
⇒ 1 = a + b + 2 [∵ f(1) = 1 & c = 2]
⇒ a + b = 1 – 2
⇒ a + b = -1 …(iii)
Finally, replace x by 4 in equation (i), we get
f(4) = a(4)2 + b(4) + c
⇒ 6 = 16a + 4b + 2 [∵ f(4) = 6 & c = 2]
⇒ 16a + 4b = 6 – 2
⇒ 16a + 4b = 4
⇒ 4 (4a + b) = 4
⇒ 4a + b = 1 …(iv)
Solving equations (iii) & (iv), we get
⇒ -3a = -2
⇒ 
Put a = 2/3 in equation (iii), we get
⇒ 
⇒ 
⇒ 
Thus, a = 2/3, b = -5/3 and c = 2.
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