To find the roots of the function gx = x² + 3x + 4x² + 4x + 29, we first need to simplify the expression.
Combine like terms:
- 4x² + x² = 5x²
- 3x + 4x = 7x
This gives us the simplified function:
gx = 5x² + 7x + 29
Next, we need to determine the roots of this quadratic equation.
We can use the quadratic formula:
x = (-b ± √(b² – 4ac)) / 2a
Here, a = 5, b = 7, and c = 29.
Now, we calculate the discriminant, b² – 4ac:
Discriminant = 7² – 4 * 5 * 29
Calculating this gives:
Discriminant = 49 – 580 = -531
Since the discriminant is negative, it indicates that the quadratic equation has no real roots. Instead, it has two complex roots.
We can further calculate the complex roots.
- The real part will be -b/(2a) = -7/(2*5) = -7/10
- The imaginary part will be √(-531)/(2*5) = √(531)i/10
Thus, the roots of the function gx are:
- x = -7/10 + √531i/10
- x = -7/10 – √531i/10
In conclusion, the roots of the equation gx = x² + 3x + 4x² + 4x + 29 are complex numbers due to the negative discriminant.