Question
Find the discriminant.
2z^2 + 5z + 2 = 0
How many real solutions does the equation have?
2z^2 + 5z + 2 = 0
How many real solutions does the equation have?
Answers
GPT-4o mini
To find the discriminant of the quadratic equation \(2z^2 + 5z + 2 = 0\), we can use the formula for the discriminant \(D\):
\[
D = b^2 - 4ac
\]
where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation \(az^2 + bz + c = 0\).
In this equation, we have:
- \(a = 2\)
- \(b = 5\)
- \(c = 2\)
Now, we can substitute these values into the discriminant formula:
\[
D = (5)^2 - 4(2)(2)
\]
\[
D = 25 - 16
\]
\[
D = 9
\]
The discriminant \(D = 9\) is greater than zero.
When the discriminant is positive, it indicates that the quadratic equation has two distinct real solutions.
Thus, the equation \(2z^2 + 5z + 2 = 0\) has **two distinct real solutions**.
\[
D = b^2 - 4ac
\]
where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation \(az^2 + bz + c = 0\).
In this equation, we have:
- \(a = 2\)
- \(b = 5\)
- \(c = 2\)
Now, we can substitute these values into the discriminant formula:
\[
D = (5)^2 - 4(2)(2)
\]
\[
D = 25 - 16
\]
\[
D = 9
\]
The discriminant \(D = 9\) is greater than zero.
When the discriminant is positive, it indicates that the quadratic equation has two distinct real solutions.
Thus, the equation \(2z^2 + 5z + 2 = 0\) has **two distinct real solutions**.