Question
Find the discriminant.
4z^2 + 4z + 1 = 0
How many real solutions does the equation have?
4z^2 + 4z + 1 = 0
How many real solutions does the equation have?
Answers
GPT-4o mini
To find the discriminant of the quadratic equation \(4z^2 + 4z + 1 = 0\), we can use the formula for the discriminant \(D\) of a quadratic equation in the form \(az^2 + bz + c = 0\), which is given by:
\[
D = b^2 - 4ac
\]
In our equation, \(a = 4\), \(b = 4\), and \(c = 1\).
Now, we can calculate the discriminant:
\[
D = (4)^2 - 4(4)(1)
\]
\[
D = 16 - 16
\]
\[
D = 0
\]
The discriminant \(D = 0\) indicates that the equation has exactly one real solution (or a repeated real solution).
Thus, the equation \(4z^2 + 4z + 1 = 0\) has one real solution.
\[
D = b^2 - 4ac
\]
In our equation, \(a = 4\), \(b = 4\), and \(c = 1\).
Now, we can calculate the discriminant:
\[
D = (4)^2 - 4(4)(1)
\]
\[
D = 16 - 16
\]
\[
D = 0
\]
The discriminant \(D = 0\) indicates that the equation has exactly one real solution (or a repeated real solution).
Thus, the equation \(4z^2 + 4z + 1 = 0\) has one real solution.