Question

solving a polynomial inequality given a graph

consider the following quadratic function and its graph.
$f(x) = x^2 - 4x - 21$
graph of the quadratic function (parabola) on a grid with $x$-axis from $-16$ to $16$ and $y$-axis from $-16$ to $16$

what is the solution of the following quadratic inequality?
$x^2 - 4x - 21 \leq 0$
multiple-choice options: number lines with $x$-axis from $-8$ to $8$, marked with open/closed circles at various points

Explanation:

Step1: Find roots of the quadratic equation

To solve \(x^{2}-4x - 21=0\), we factor the quadratic. We need two numbers that multiply to \(- 21\) and add to \(-4\). The numbers are \(-7\) and \(3\). So, \(x^{2}-4x - 21=(x - 7)(x+3)=0\). Setting each factor equal to zero gives \(x - 7 = 0\) or \(x+3=0\), so the roots are \(x = 7\) and \(x=-3\).

Step2: Analyze the graph of the quadratic function

The quadratic function \(f(x)=x^{2}-4x - 21\) has a positive leading coefficient (\(a = 1>0\)), so its graph is a parabola opening upwards. The inequality \(x^{2}-4x - 21\leqslant0\) represents the region where the graph of the function is below or on the \(x -\)axis. For a parabola opening upwards, the function is less than or equal to zero between its two roots (including the roots).

Step3: Determine the solution interval

Since the roots are \(x=-3\) and \(x = 7\), the solution to \(x^{2}-4x - 21\leqslant0\) is \(-3\leqslant x\leqslant7\). Looking at the number lines, the one with closed circles at \(-3\) (or \(- 3\) equivalent on the number line, here the second option has closed circles at \(-3\) and \(7\) - like points, and the line between them is shaded) represents the interval \([-3,7]\).

Answer:

The number line with closed circles (filled dots) at \(-3\) (or the left closed point) and \(7\) (or the right closed point) and the line segment between them shaded (the second option among the given number lines). In interval notation, the solution is \(\boldsymbol{[-3,7]}\)