When does the function gx = 6x² + 23x + 4 equal zero?

To determine when the function g(x) = 6x² + 23x + 4 equals zero, we need to find the roots of the equation. This means solving the equation:

6x² + 23x + 4 = 0

We can apply the quadratic formula, which is given by:

x = (-b ± √(b² – 4ac)) / 2a

In our case:

• a = 6
• b = 23
• c = 4

Now, substituting the values into the formula:

x = ( -23 ± √(23² - 4 * 6 * 4) ) / (2 * 6)

First, let’s calculate the discriminant:

23² - 4 * 6 * 4 = 529 - 96 = 433

The discriminant is positive (433), which means there are two real roots. Now we can proceed with the calculations:

x = ( -23 ± √433 ) / 12

This gives us two solutions:

• x₁ = ( -23 + √433) / 12
• x₂ = ( -23 – √433) / 12

Calculating these values:

x₁ ≈ -0.215

and

x₂ ≈ -3.785

Thus, the values of x where the function g(x) equals zero are:

• x ≈ -0.215
• x ≈ -3.785

In summary, the function g(x) = 6x² + 23x + 4 equals zero at approximately -0.215 and -3.785.