To solve the equation 4x² + 3x + 9 = 2x + 1, we first rearrange it into standard quadratic form:
4x² + 3x + 9 – 2x – 1 = 0
This simplifies to:
4x² + (3x – 2x) + (9 – 1) = 0
or:
4x² + x + 8 = 0
Now, we can identify the coefficients for the quadratic formula, which is:
x = (-b ± √(b² – 4ac)) / 2a
For our equation, the coefficients are:
- a = 4
- b = 1
- c = 8
Next, we calculate the discriminant, b² – 4ac:
Discriminant = 1² – 4(4)(8) = 1 – 128 = -127
Since the discriminant is negative, it indicates that there are no real solutions; the solutions will be complex numbers.
Now, we can proceed to find the complex solutions using the quadratic formula:
x = (-1 ± √(-127)) / (2 * 4)
This can be rewritten as:
x = (-1 ± i√127) / 8
Thus, the values of x are:
- x = (-1 + i√127) / 8
- x = (-1 – i√127) / 8
In summary, we found that the values of x are complex and can be expressed as:
x = (-1 ± i√127) / 8