To solve the quadratic inequality (3x^2 - 5x + 4 < 0), we first need to analyze the quadratic expression.
Step 1: Find the roots of the quadratic equation
We can find the roots of the corresponding quadratic equation (3x^2 - 5x + 4 = 0) using the quadratic formula:
[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
]
where (a = 3), (b = -5), and (c = 4).
Step 2: Calculate the discriminant
First, calculate the discriminant ((D)):
[
D = b^2 - 4ac = (-5)^2 - 4(3)(4) = 25 - 48 = -23
]
Step 3: Interpret the discriminant
Since the discriminant is negative ((D < 0)), this means that the quadratic equation (3x^2 - 5x + 4 = 0) has no real roots. As a result, the quadratic function does not cross the x-axis.
Step 4: Determine the sign of the quadratic expression
Next, since the leading coefficient (a = 3) is positive, the quadratic (3x^2 - 5x + 4) opens upwards. Therefore, the entire quadratic expression will be greater than zero for all (x).
Conclusion
The inequality (3x^2 - 5x + 4 < 0) has no solutions, since there are no x-values for which the quadratic expression is negative.
In summary:
[
\text{The solution to } 3x^2 - 5x + 4 < 0 \text{ is } \text{no solutions}.
]
