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 \(D\):
- If \(D > 0\): There are two distinct real solutions.
- If \(D = 0\): There is exactly one real solution (a repeated root).
- 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 \(x^2 - 5x + 4 = 0\).
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