To solve the problem, we start with the given quadratic expression:
[tex]\[ f(x) = ax^2 + bx \][/tex]
We are given two conditions:
1. [tex]\( f(7) = f(-2) \)[/tex]
2. [tex]\( f(6) = 12 \)[/tex]
### Step 1: Apply the First Condition (f(7) = f(-2))
Let's compute [tex]\( f(7) \)[/tex] and [tex]\( f(-2) \)[/tex]:
[tex]\[ f(7) = a(7)^2 + b(7) = 49a + 7b \][/tex]
[tex]\[ f(-2) = a(-2)^2 + b(-2) = 4a - 2b \][/tex]
Setting these equal per the given condition:
[tex]\[ 49a + 7b = 4a - 2b \][/tex]
Rearrange this equation to isolate variables:
[tex]\[ 49a + 7b - 4a + 2b = 0 \][/tex]
[tex]\[ 45a + 9b = 0 \][/tex]
Divide the entire equation by 9:
[tex]\[ 5a + b = 0 \][/tex]
From this, we can solve for [tex]\( b \)[/tex]:
[tex]\[ b = -5a \][/tex]
### Step 2: Apply the Second Condition (f(6) = 12)
Now we use the second given condition:
[tex]\[ f(6) = 12 \][/tex]
Compute [tex]\( f(6) \)[/tex]:
[tex]\[ f(6) = a(6)^2 + b(6) = 36a + 6b \][/tex]
Set this equal to 12:
[tex]\[ 36a + 6b = 12 \][/tex]
Substitute [tex]\( b = -5a \)[/tex] from the first step into this equation:
[tex]\[ 36a + 6(-5a) = 12 \][/tex]
[tex]\[ 36a - 30a = 12 \][/tex]
[tex]\[ 6a = 12 \][/tex]
Solve for [tex]\( a \)[/tex]:
[tex]\[ a = 2 \][/tex]
### Step 3: Determine the Value of [tex]\( b \)[/tex]
Using the value of [tex]\( a = 2 \)[/tex], calculate [tex]\( b \)[/tex]:
[tex]\[ b = -5a \][/tex]
[tex]\[ b = -5(2) \][/tex]
[tex]\[ b = -10 \][/tex]
Thus, the value of [tex]\( b \)[/tex] in the quadratic expression is [tex]\( \boxed{-10} \)[/tex].
