We are given a quadratic function $g(x) = ax^2 + bx + 1$. We are also given that $g(1) = 0$ and $g(2) = 3$. We need to find two equations in terms of $a$ and $b$, and then solve these equations to find the values of $a$ and $b$.

1. Problem Description

We are given a quadratic function $g(x) = ax^2 + bx + 1$. We are also given that $g(1) = 0$ and $g(2) = 3$. We need to find two equations in terms of $a$ and $b$, and then solve these equations to find the values of $a$ and $b$.

2. Solution Steps

First, use the given information $g(1) = 0$:

$g(1) = a(1)^2 + b(1) + 1 = 0$

$a + b + 1 = 0$

$a + b = -1$ (Equation 1)

Next, use the given information $g(2) = 3$:

$g(2) = a(2)^2 + b(2) + 1 = 3$

$4a + 2b + 1 = 3$

$4a + 2b = 2$

$2a + b = 1$ (Equation 2)

Now we have two equations:

Equation 1: $a + b = -1$

Equation 2: $2a + b = 1$

We can solve this system of equations using substitution or elimination. Let's use elimination. Subtract Equation 1 from Equation 2:

$(2a + b) - (a + b) = 1 - (-1)$

$2a + b - a - b = 2$

$a = 2$

Now substitute $a = 2$ into Equation 1:

$2 + b = -1$

$b = -1 - 2$

$b = -3$

So, $a = 2$ and $b = -3$.

3. Final Answer

$a = 2$ and $b = -3$.