Question

consider the following quadratic function and its graph. (f(x)=x^{2}-4x - 21). what is the solution of the following quadratic inequality? (x^{2}-4x - 21leq0)

Explanation:

Step1: Factor the quadratic expression

$x^{2}-4x - 21=(x - 7)(x+3)$

Step2: Find the roots of the quadratic equation

Set $(x - 7)(x + 3)=0$. Then $x-7=0$ gives $x = 7$ and $x+3=0$ gives $x=-3$.

Step3: Determine the solution of the inequality

We want to find when $(x - 7)(x + 3)\leq0$. The quadratic function $y=(x - 7)(x + 3)$ is a parabola opening upwards (since the coefficient of $x^{2}$ is positive). The inequality is satisfied when $-3\leq x\leq7$. On a number - line, this is represented by a closed - circle at $x=-3$ and $x = 7$ and the line segment between them.

Answer:

The second option (the number - line with closed circles at $x=-3$ and $x = 7$ and the line segment between them)