The-quadratic-polynomials-p-x-a-x-3-2-bx-1-and-q-x-2x-2-c-x-2-13-are-equal-for-all-values-of-x-Find-the-values-of-a-b-and-c-

Question Number 16881 by Tinkutara last updated on 27/Jun/17

The quadratic polynomials

p (x) = a {(x - 3)}^{2} + b x + 1 and

q (x) = 2 x^{2} + c (x - 2) + 13 are equal

for all values of x . Find the values of a,

b, and c .

Answered by RasheedSoomro last updated on 27/Jun/17

p (0) = q (0) \Rightarrow 9 a + 1 = - 2 c + 13

9 a + 2 c = 12 \dots \dots \dots (i)

p (3) = q (3) \Rightarrow 3 b + 1 = 18 + c + 13

3 b - c = 30 \dots \dots \dots . (ii)

p (2) = q (2) \Rightarrow a + 2 b + 1 = 8 + 13

a + 2 b = 20 \dots \dots \dots . . (iii)

(ii) \Rightarrow c = 3 b - 30

(i) \Rightarrow 9 a + 2 (3 b - 30) = 12 \Rightarrow 3 a + 2 b = 24 \dots \dots \dots (iv)

(iii), (iv) \Rightarrow 3 (20 - 2 b) + 2 b = 24 \Rightarrow - 4 b = 24 - 60

b = \frac{- 36}{- 4} = 9

a = 20 - 2 (9) = 2

c = 3 (9) - 30 = - 3

a = 2, b = 9, c = - 3

Commented by Tinkutara last updated on 28/Jun/17

Thanks Sir!

Answered by RasheedSoomro last updated on 28/Jun/17

AnOther way

p (x) = a (x^{2} - 6 x + 9) + bx + 1

= {ax}^{2} - 6 ax + 9 a + bx + 1

= {ax}^{2} + (- 6 a + b) x + 9 a + 1

q (x) = {2 x}^{2} + cx - 2 c + 13

Comparing coefficients of x

a = 2 \land - 6 a + b = c \land 9 a + 1 = - 2 c + 13

- 6 (2) + b = c \land 9 (2) + 1 = - 2 c + 13

b - c = 12 \land c = - 3

b - (- 3) = 12

b = 9