Problem 40 - Algebra and Trigonometry

Question

Find the quadratic function \(f(x)=a x^{2}+b x+c\) for which \(f(-1)=5, f(1)=3,\) and \(f(2)=5\)

Step-by-Step Solution

Answer

The quadratic function satisfying the given conditions is \(f(x) = x^2 - x + 3\).

Step 1: Set up system of equations

Using the expressions given \(f(-1)=5, f(1)=3,\) and \(f(2)=5\), create three equations. Plug \(-1\), \(1\), and \(2\) into the quadratic function \(f(x)=a*x^2+b*x+c\) and equate them to \(5\), \(3\), and \(5\) respectively. The resulting equations are: \[ a - b + c = 5 \] \[ a + b + c = 3 \] \[ 4a + 2b + c = 5 \]

Step 2: Solve system of equations

Solve these three equations simultaneously using any of the methods for solving linear equations. Subtraction of the first two equations gives \(2b = -2\), which can be simplified to \(b = -1\). Replacing \(b\) with \(-1\) in the first equation leads to \(a - (-1) + c = 5\), which simplifies to \(a + c = 4\). Using \(b = -1\) in the third equation yields \(4a - 2 + c = 5\), which can be further simplified to \(4a + c = 7\). Thus the system of equations to solve is now: \[ a + c = 4 \] \[ 4a + c = 7 \]

Step 3: Find a, b, and c

Solving the last two equations gives \(a = 1\) and \(c = 3\). Thus, the original function \(f(x) = ax^2 + bx + c\) with \(a = 1\), \(b = -1\), and \(c = 3\), can be expressed as: \[f(x) = x^2 - x + 3\].