SOLUTION: A function is defined by g:x=ax²+bx+2 where a and b are constants. If g(-2)=6 and g(3)=11, Find the values of a and b

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Question 1196172: A function is defined by g:x=ax²+bx+2 where a and b are constants. If g(-2)=6 and g(3)=11, Find the values of a and b
Found 4 solutions by MathLover1, josgarithmetic, math_tutor2020, MathTherapy:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

A function is defined by:
g%28x%29=ax%5E2%2Bbx%2B2 where a and+b are constants

If g%28-2%29=6+
6=a%28-2%29%5E2%2Bb%28-2%29%2B2
6=4a-2b%2B2............solve for a
2b%2B6-2=4a
2b%2B4=4a ............simplify
b%2B2=2a+
a=+b%2F2%2B1...............eq.1


and if g%283%29=11, we have

11=a%283%29%5E2%2Bb%283%29%2B2
11=9a%2B3b%2B2
11-2-3b=9a+
9-3b=9a+
a=9%2F9-3b%2F9
a=1-b%2F3................eq.2

from eq.1 and eq.2 we have

b%2F2%2B1=1-b%2F3
b%2F2%2Bb%2F3=1-1
b%2F2%2Bb%2F3=0 -> will be true only if+b=0

go to a=1-b%2F3................eq.2, plug in b=0 and we get
a=1
the values of a+and b are 1+and 0+respectively

and your function is:
g%28x%29=1%2Ax%5E2%2B0%2Ax%2B2+
g%28x%29=x%5E2%2B2+



Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
Maybe you mean g%28x%29=ax%5E2%2Bbx%2B2, and for known points (-2,6) and (3,11) ?


Using the given points,
system%284a-2b%2B2=6%2C9a%2B3b%2B2=11%29

and steps to solve for a and b,
system%284a-2b=4%2C9a%2B3b=9%29

system%282a-b=2%2C3a%2Bb=3%29

Add corresponding members:
5a=5
highlight%28a=1%29
and you can easily solve for b,.....

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

Plug in x = -2
g(x) = a*x^2 + b*x + 2
g(-2) = a(-2)^2 + b(-2) + 2
g(-2) = a(4) + b(-2) + 2
g(-2) = 4a - 2b + 2
6 = 4a - 2b + 2
6-2 = 4a - 2b
4 = 4a - 2b

Let's solve for b
4 = 4a - 2b
4+2b = 4a
2b = 4a-4
b = 4a/2-4/2
b = 2a - 2

Now plug in x = 3
g(x) = ax^2 + bx + 2
g(3) = a(3)^2 + b(3) + 2
11 = 9a + 3b + 2

From here, plug in b = 2a - 2 and solve for 'a'
11 = 9a + 3b + 2
11 = 9a + 3(2a-2) + 2
11 = 9a + 6a-6 + 2
11 = 15a-4
15a-4 = 11
15a = 11+4
15a = 15
a = 15/15
a = 1

Then,
b = 2a-2
b = 2(1)-2
b = 2-2
b = 0


Answers:
a = 1
b = 0
The function is g(x) = x^2+2

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!
A function is defined by g:x=ax²+bx+2 where a and b are constants. If g(-2)=6 and g(3)=11, Find the values of a and b
I hope for your sanity that you didn't pay any attention to that woman's method of solving this porblem. She has this habit
of solving a variable in terms of another variable in fractional form, which is TOTALLY UNNECESSARY and LUDICROUS!! 

g(-2) = 6 signifies that x = - 2 when y = 6. We then have: 
   matrix%281%2C3%2C+g%28x%29%2C+%22=%22%2C+ax%5E2+%2B+bx+%2B+2%29
 matrix%281%2C3%2C+g%28-+2%29%2C+%22=%22%2C+a%28-+2%29%5E2+%2B+b%28-+2%29+%2B+2%29 ---- Substituting - 2 for x
g(- 2) = 4a - 2b + 2
     6 = 4a - 2b + 2 ----- Substituting 6 for y, or g(- 2)  
     4 = 4a - 2b____2(2) = 2(2a - b)_____2 = 2a - b ----- eq (i)

Likewise,  g(3) = 11 signifies that x = 3 when y = 11. We then have: 
 matrix%281%2C3%2C+g%28x%29%2C+%22=%22%2C+ax%5E2+%2B+bx+%2B+2%29
 matrix%281%2C3%2C+g%283%29%2C+%22=%22%2C+a%283%29%5E2+%2B+b%283%29+%2B+2%29 ----- Substituting 3 for x
 g(3) = 9a + 3b + 2
   11 = 9a + 3b + 2 ----- Substituting 11 for y, or g(3)  
    9 = 9a + 3b____3(3) = 3(3a + b)_____3 = 3a + b ----- eq (ii)

We then have:
2 = 2a - b ----- eq (i)
3 = 3a + b ----- eq (ii
5 = 5a ---- Adding eqs (i) & (ii)


3 = 3(1) + b ------ Substituting 1 for a in eq (ii)