Answer :
To find the values of the constants \( p \), \( q \), and \( r \) in the function \( f(x) = p x^2 + q x + r \) given the conditions \( f(0) = 4 \), \( f(-1) = 6 \), and \( f(-2) = 18 \), we follow these steps:
1. Substitute \( x = 0 \) into the function:
[tex]\[ f(0) = p(0)^2 + q(0) + r = r \][/tex]
We know \( f(0) = 4 \), so:
[tex]\[ r = 4 \][/tex]
2. Substitute \( x = -1 \) into the function:
[tex]\[ f(-1) = p(-1)^2 + q(-1) + r = p(1) - q + r = p - q + r \][/tex]
We know \( f(-1) = 6 \), so:
[tex]\[ p - q + r = 6 \][/tex]
Substituting \( r = 4 \) from the first condition:
[tex]\[ p - q + 4 = 6 \implies p - q = 2 \quad \text{(Equation 1)} \][/tex]
3. Substitute \( x = -2 \) into the function:
[tex]\[ f(-2) = p(-2)^2 + q(-2) + r = p(4) - 2q + r = 4p - 2q + r \][/tex]
We know \( f(-2) = 18 \), so:
[tex]\[ 4p - 2q + r = 18 \][/tex]
Substituting \( r = 4 \) from the first condition:
[tex]\[ 4p - 2q + 4 = 18 \implies 4p - 2q = 14 \quad \text{(Equation 2)} \][/tex]
4. Solve the system of equations derived from steps 2 and 3:
From Equation 1:
[tex]\[ p - q = 2 \][/tex]
From Equation 2:
[tex]\[ 4p - 2q = 14 \][/tex]
We can solve these equations simultaneously. First, solve Equation 1 for \( p \):
[tex]\[ p = q + 2 \][/tex]
Substitute \( p = q + 2 \) into Equation 2:
[tex]\[ 4(q + 2) - 2q = 14 \][/tex]
Simplify and solve for \( q \):
[tex]\[ 4q + 8 - 2q = 14 \implies 2q + 8 = 14 \implies 2q = 6 \implies q = 3 \][/tex]
5. Find \( p \) using the value of \( q \):
[tex]\[ p = q + 2 = 3 + 2 = 5 \][/tex]
Hence, the values of the constants are:
[tex]\[ p = 5, \quad q = 3, \quad r = 4 \][/tex]
1. Substitute \( x = 0 \) into the function:
[tex]\[ f(0) = p(0)^2 + q(0) + r = r \][/tex]
We know \( f(0) = 4 \), so:
[tex]\[ r = 4 \][/tex]
2. Substitute \( x = -1 \) into the function:
[tex]\[ f(-1) = p(-1)^2 + q(-1) + r = p(1) - q + r = p - q + r \][/tex]
We know \( f(-1) = 6 \), so:
[tex]\[ p - q + r = 6 \][/tex]
Substituting \( r = 4 \) from the first condition:
[tex]\[ p - q + 4 = 6 \implies p - q = 2 \quad \text{(Equation 1)} \][/tex]
3. Substitute \( x = -2 \) into the function:
[tex]\[ f(-2) = p(-2)^2 + q(-2) + r = p(4) - 2q + r = 4p - 2q + r \][/tex]
We know \( f(-2) = 18 \), so:
[tex]\[ 4p - 2q + r = 18 \][/tex]
Substituting \( r = 4 \) from the first condition:
[tex]\[ 4p - 2q + 4 = 18 \implies 4p - 2q = 14 \quad \text{(Equation 2)} \][/tex]
4. Solve the system of equations derived from steps 2 and 3:
From Equation 1:
[tex]\[ p - q = 2 \][/tex]
From Equation 2:
[tex]\[ 4p - 2q = 14 \][/tex]
We can solve these equations simultaneously. First, solve Equation 1 for \( p \):
[tex]\[ p = q + 2 \][/tex]
Substitute \( p = q + 2 \) into Equation 2:
[tex]\[ 4(q + 2) - 2q = 14 \][/tex]
Simplify and solve for \( q \):
[tex]\[ 4q + 8 - 2q = 14 \implies 2q + 8 = 14 \implies 2q = 6 \implies q = 3 \][/tex]
5. Find \( p \) using the value of \( q \):
[tex]\[ p = q + 2 = 3 + 2 = 5 \][/tex]
Hence, the values of the constants are:
[tex]\[ p = 5, \quad q = 3, \quad r = 4 \][/tex]
