Question
x^2 - 5x + 4=0
Compute the discriminant. Then determine the number and type of solutions for the given equation.
Compute the discriminant. Then determine the number and type of solutions for the given equation.
Answers
GPT-4o mini
To determine the discriminant of the quadratic equation \(x^2 - 5x + 4 = 0\), we can use the formula for the discriminant of a quadratic equation \(ax^2 + bx + c = 0\), which is given by:
\[
D = b^2 - 4ac
\]
In our equation, we can identify the coefficients as follows:
- \(a = 1\)
- \(b = -5\)
- \(c = 4\)
Now we can substitute these values into the discriminant formula:
\[
D = (-5)^2 - 4(1)(4)
\]
\[
D = 25 - 16
\]
\[
D = 9
\]
Now that we have computed the discriminant, we can analyze the number and type of solutions based on the value of \(D\):
1. If \(D > 0\): There are two distinct real solutions.
2. If \(D = 0\): There is exactly one real solution (a repeated root).
3. If \(D < 0\): There are no real solutions (the solutions are complex).
Since we found that \(D = 9\), which is greater than 0, we conclude that there are two distinct real solutions for the equation \(x^2 - 5x + 4 = 0\).
\[
D = b^2 - 4ac
\]
In our equation, we can identify the coefficients as follows:
- \(a = 1\)
- \(b = -5\)
- \(c = 4\)
Now we can substitute these values into the discriminant formula:
\[
D = (-5)^2 - 4(1)(4)
\]
\[
D = 25 - 16
\]
\[
D = 9
\]
Now that we have computed the discriminant, we can analyze the number and type of solutions based on the value of \(D\):
1. If \(D > 0\): There are two distinct real solutions.
2. If \(D = 0\): There is exactly one real solution (a repeated root).
3. If \(D < 0\): There are no real solutions (the solutions are complex).
Since we found that \(D = 9\), which is greater than 0, we conclude that there are two distinct real solutions for the equation \(x^2 - 5x + 4 = 0\).