To find the discriminant of the quadratic equation \(2w^2 + 4w + 2 = 0\), we can use the formula for the discriminant, \(D\), which is given by:
\[ D = b^2 - 4ac \]
where \(a\), \(b\), and \(c\) are the coefficients from the standard form of a quadratic equation \(ax^2 + bx + c = 0\).
For the equation \(2w^2 + 4w + 2 = 0\):
- \(a = 2\)
- \(b = 4\)
- \(c = 2\)
Now, we can substitute these values into the discriminant formula:
\[ D = (4)^2 - 4(2)(2) \]
Calculating this gives:
\[ D = 16 - 16 \] \[ D = 0 \]
The discriminant \(D\) is \(0\).
When the discriminant is:
- \(D > 0\): there are two distinct real solutions.
- \(D = 0\): there is exactly one real solution (a repeated root).
- \(D < 0\): there are no real solutions.
Since the discriminant is \(0\), the equation \(2w^2 + 4w + 2 = 0\) has exactly one real solution.