The expression 3x^2-x-9 is rearranged into the form a(x+2)^2+b(x+2)+c. What are the values of a,b,and c?
3x^2-x-9=a(x+2)^2+b(x+2)+c
compare coefficent
now
a=?
b=?
c=?
ooh yes yes very correct i know that
To rearrange the expression 3x^2 - x - 9 into the form a(x + 2)^2 + b(x + 2) + c, we can complete the square.
Step 1: Factor out the coefficient of the x^2 term:
3x^2 - x - 9 = 3(x^2 - (1/3)x) - 9
Step 2: Calculate half the coefficient of the x term and square it:
(1/2)(-1/3) = -1/6
(-1/6)^2 = 1/36
Step 3: Add and subtract the value from step 2 inside the brackets, while taking the multiplied value outside the bracket:
3(x^2 - (1/3)x + 1/36 - 1/36) - 9 = 3((x - 1/6)^2 - 1/36) - 9
Expand the square inside the bracket:
3(x - 1/6)^2 - 3/36 - 9 = 3(x - 1/6)^2 - 1/12 - 9
Combine constants:
-1/12 - 9 = -1/12 - 108/12 = -109/12
Now we have the expression in the required form: a(x + 2)^2 + b(x + 2) + c
Comparing the expression above to the rearranged expression, a = 3, b = -109/12, and c = -1/12.
You would first of all have to expand
a(x+2)^2 + b(x+2) + c
= a(x^2 + 4x + 4) + bx + 2b + c
= ax^2 + 4ax + 4a + bx + 2b + c
= ax^2 + (4a+b)x + (4a + 2b + c)
now compare:
3x^2 <---> ax^2 ---- a = 3
4a+b)x <---> -x
4a + b = -1
12+b = -1
b = -13
4a+2b+c = -9
12 - 26 + c = -9
c = 5
check:
3(x+2)^2 - 13(x+2) + 5
= 3(x^2 + 4x + 4) - 13x - 26 + 5
= 3x^2 + 12x + 12 - 13x - 26 + 5
= 3x^2 - x - 9 , as required