To determine when the function gx = 6x² + 23x + 4 equals zero, we need to solve the equation:
6x² + 23x + 4 = 0
This is a quadratic equation in the standard form of ax² + bx + c = 0, where a = 6, b = 23, and c = 4. We can use the quadratic formula:
x = (-b ± √(b² – 4ac)) / (2a)
First, we calculate the discriminant:
b² – 4ac = 23² – 4(6)(4) = 529 – 96 = 433
Since the discriminant is positive, we will have two real and distinct solutions. Now, we can substitute these values into the quadratic formula:
x = ( -23 ± √433 ) / (12)
This gives us the two solutions:
x₁ = ( -23 + √433 ) / (12)
x₂ = ( -23 – √433 ) / (12)
Thus, the function gx equals zero for these two values of x.