How many \( x \)-intercepts does the parabola with the following equation have? \( y=2 x^{2}+14 x+22 \) Show your work here Hint To add infinity (0), type "infinity"

To determine the number of \( x \)-intercepts of the parabola given by the equation \[ y = 2x^2 + 14x + 22, \] we need to find the values of \( x \) for which \( y = 0 \). This means we need to solve the equation: \[ 2x^2 + 14x + 22 = 0. \] To find the number of solutions to this quadratic equation, we can use the discriminant, which is given by the formula: \[ D = b^2 - 4ac, \] where \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation \( ax^2 + bx + c = 0 \). In our case, we have: - \( a = 2 \) - \( b = 14 \) - \( c = 22 \) Now, we can calculate the discriminant: \[ D = (14)^2 - 4(2)(22). \] Calculating \( (14)^2 \): \[ 14^2 = 196. \] Now calculating \( 4(2)(22) \): \[ 4 \cdot 2 \cdot 22 = 176. \] Now substituting these values into the discriminant formula: \[ D = 196 - 176 = 20. \] Since the discriminant \( D \) is positive (\( D > 0 \)), this indicates that the quadratic equation has two distinct real solutions. Therefore, the parabola intersects the \( x \)-axis at two points. Thus, the number of \( x \)-intercepts of the parabola is \[ \boxed{2}. \]