To solve the equation 4x2 + 3x + 9 = 2x + 1 using the quadratic formula, we first need to rearrange the equation into standard quadratic form, which is ax2 + bx + c = 0.
1. Start by moving all terms to one side of the equation:
4x2 + 3x + 9 – 2x – 1 = 0
2. Combine like terms:
4x2 + (3x – 2x) + (9 – 1) = 0
which simplifies to:
4x2 + x + 8 = 0
Now we identify the coefficients a, b, and c:
- a = 4
- b = 1
- c = 8
3. Next, we apply the quadratic formula, which is given by:
x = (-b ± √(b² – 4ac)) / 2a
4. Substitute the values of a, b, and c into the formula:
x = (-(1) ± √((1)² – 4 * (4) * (8))) / (2 * 4)
5. Calculate the discriminant:
(1)² – 4 * 4 * 8 = 1 – 128 = -127
Since the discriminant is negative, this means the solutions will be complex numbers. Thus:
6. Proceed with the calculation:
x = (-1 ± √(-127)) / 8
We can express the square root of a negative number using the imaginary unit i
x = (-1 ± i√127) / 8
7. Finally, we can write the solutions as:
x = -1/8 ± i√127/8
In conclusion, the values of x are:
- x = -1/8 + i√127/8
- x = -1/8 – i√127/8
These represent the complex solutions to the given quadratic equation.