To solve the equation 7x² + 7 = 0 using the quadratic formula, we first need to rewrite it in the standard form of a quadratic equation, which is ax² + bx + c = 0.
In our case, we can identify:
- a = 7
- b = 0
- c = 7
The quadratic formula is given by:
x = (-b ± √(b² – 4ac)) / (2a)
Now, we can substitute the values of a, b, and c into the formula:
x = (0 ± √(0² – 4 * 7 * 7)) / (2 * 7)
This simplifies to:
x = (0 ± √(0 – 196)) / 14
Which further simplifies to:
x = (0 ± √(-196)) / 14
Because the square root of a negative number involves imaginary numbers, we can express this as:
x = (0 ± 14i) / 14
Dividing through gives us:
x = ± i
Thus, the values of x are:
x = i (the positive imaginary unit) and x = -i (the negative imaginary unit).
These results indicate that the solutions to the equation are purely imaginary numbers. Therefore, the equation 7x² + 7 = 0 does not have any real solutions.