Question

solve the following inequality.
$x^2 - 3x + 8 \leq 0$
select the correct choice below, and if necessary, fill in the answer box to complete your choice.
\\(\bigcirc\\) a. the solution is \\(\square\\)
(type your answer in interval notation.)
\\(\bigcirc\\) b. there is no real solution.

Explanation:

Step1: Analyze the quadratic function

For the quadratic inequality \(x^2 - 3x + 8 \leq 0\), consider the quadratic function \(f(x)=x^2 - 3x + 8\). The discriminant of a quadratic \(ax^2+bx+c\) is \(\Delta = b^2 - 4ac\). Here, \(a = 1\), \(b=-3\), \(c = 8\), so \(\Delta=(-3)^2-4\times1\times8=9 - 32=-23\).

Step2: Determine the graph of the function

Since \(a = 1>0\), the parabola \(y = f(x)\) opens upwards. And since \(\Delta=-23<0\), the parabola has no real roots and is always above the \(x\)-axis (because it never intersects the \(x\)-axis and opens upwards). So \(f(x)=x^2 - 3x + 8>0\) for all real \(x\), meaning \(x^2 - 3x + 8 \leq 0\) has no real solutions.

Answer:

B. There is no real solution.