Please Urgent Help Needed (1 Viewer)

[keetviews]

Find a, b and c, given that the following pairs of polynomials are identically equal.

a(x + 2)^2 + b(x + 3)^2 + c(x + 4)^2 = 2x^2 + 8x + 6, for all x.

Please I need help with finding the coefficients of these types of questions.

 

[Eagle Mum]

a(x + 2)^2 + b(x + 3)^2 + c(x + 4)^2 = 2x^2 + 8x + 6

Expand LHS:
ax^2 + 4ax + 4a + bx^2 + 6bx + 9b + cx^2 + 8cx + 16c = 2x^2 + 8x + 6

Group the coefficients of the same powers of x on LHS:
(a+b+c)x^2 + (4a+6b+8c)x + (4a+9b+16c) = 2x^2 + 8x + 6

Match coefficients of the LHS & RHS to form three simultaneous equations:

1. a+b+c = 2
2. 4a+6b+8c = 8
3. 4a+9b+16c = 6

Multiply eqn 1 by 4:
4. 4a + 4b + 4c = 8

Subtract eqn 4 from eqn 2:
5. 2b+4c = 0 or b=-2c

Substitute eqn 5 in eqn 1:
6. a-2c+c = 2 or a-c = 2 or a = c+2

Substitute 5 & 6 in 3:
4c+8-18c+16c = 6 or 2c =-2

Therefore c=-1, b=2, a=1

 

[quickoats]

Alternately, if the polynomials are identically equal, they will yield the same function value for every value of x. You can then sub in certain values of x which may make the coefficients a bit easier to find.

 

[keetviews]

Thank you guys so much, this really simplifies the process!