A function of the form \( p(x)=a x^{2}+q \) has two \( x \)-intercepts, one of which is 9 . Determine the other \( x \)-intercept and explain how you arrived at your answer. \( p(x)=a x^{2}+q \)

To find the other \( x \)-intercept of the function \( p(x) = ax^2 + q \), we can follow these steps: 1. **Understanding the x-intercepts**: The \( x \)-intercepts of a function occur where \( p(x) = 0 \). Therefore, we need to solve the equation \( ax^2 + q = 0 \). 2. **Using the given information**: We know that one of the \( x \)-intercepts is 9. This means that when \( x = 9 \), \( p(9) = 0 \). We can substitute this into the equation: \[ a(9^2) + q = 0 \] Simplifying this gives: \[ 81a + q = 0 \quad \text{(1)} \] 3. **Finding the other x-intercept**: The function \( p(x) = ax^2 + q \) is a quadratic function, and it can be expressed in factored form as: \[ p(x) = a(x - r_1)(x - r_2) \] where \( r_1 \) and \( r_2 \) are the \( x \)-intercepts. Since we know one intercept \( r_1 = 9 \), we can express the function as: \[ p(x) = a(x - 9)(x - r_2) \] 4. **Finding the sum of the roots**: The sum of the roots of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( r_1 + r_2 = -\frac{b}{a} \). In our case, since there is no linear term in \( p(x) \), we can deduce that the sum of the roots must be zero (as the coefficient of \( x \) is zero). Therefore: \[ 9 + r_2 = 0 \] Solving for \( r_2 \): \[ r_2 = -9 \] Thus, the other \( x \)-intercept is \(-9\). In summary, we arrived at this conclusion by using the properties of quadratic functions and the information about the known \( x \)-intercept. The other \( x \)-intercept is \(-9\).