Certainly! Let's go through the process step-by-step to find the equation of the curve given the points where [tex]\( y = 0 \)[/tex] are [tex]\( x = -2 \)[/tex] and [tex]\( x = 3 \)[/tex].
### Step 1: Understanding the given points.
When [tex]\( y = 0 \)[/tex], the values of [tex]\( x \)[/tex] are the roots of the equation. These points are:
- [tex]\( x = -2 \)[/tex]
- [tex]\( x = 3 \)[/tex]
### Step 2: Formulating the quadratic equation using the roots.
A quadratic equation with roots [tex]\( x = -2 \)[/tex] and [tex]\( x = 3 \)[/tex] can be expressed in factored form as:
[tex]\[ y = a(x + 2)(x - 3) \][/tex]
Here, [tex]\( a \)[/tex] is a constant that will determine the shape of the parabola, but for simplicity, we can start by assuming [tex]\( a = 1 \)[/tex].
### Step 3: Expanding the factored form.
To get the quadratic function in standard form, we need to expand the equation:
[tex]\[ y = (x + 2)(x - 3) \][/tex]
### Step 4: Multiplying the binomials.
We distribute each term in the first binomial by each term in the second binomial:
[tex]\[ y = x(x - 3) + 2(x - 3) \][/tex]
[tex]\[ y = x^2 - 3x + 2x - 6 \][/tex]
### Step 5: Combining like terms.
We combine the [tex]\( x \)[/tex]-terms:
[tex]\[ y = x^2 - x - 6 \][/tex]
### Step 6: Writing the final equation.
The standard form of the quadratic equation is:
[tex]\[ y = x^2 - x - 6 \][/tex]
Therefore, the equation of the curve is:
[tex]\[ y = x^2 - x - 6 \][/tex]
And thus, the equation of the curve is:
[tex]\[ \boxed{y = x^2 - x - 6} \][/tex]
