Find the roots and use them to graph the related function.
{ x| x ≤ - 6 or x ≥ 2 }
Practice makes perfect
We will start by simplifying the inequality. Then we will sketch the related quadratic function.
Simplifying the Inequality
Note that both sides of the inequality consists of the constant terms. Let's simplify it by adding 6 to both sides of the equation.
- x^2 - 4x + 6 ≤ - 6 ⇔ - x^2 - 4x + 12 ≤ 0
Finding the Zeros
We will now find the zeros of the related quadratic equation function.
y = - x^2 - 4x + 12
To do so, we first need to identify the values of a, b, and c.
y = - x^2 - 4x + 12 ⇔ y= - 1x^2 + ( -4)x + 12
We see that a= - 1, b= - 4, and c= 12. Let's substitute these values into the Quadratic Formula to find the roots of - x^2-4x+12=0.
Now we can calculate the first root using the positive sign and the second root using the negative sign.
x=4 ± 8/- 2
x=4 + 8/- 2
x=4 - 8/- 2
x=-6
x=2
Solving the Inequality
The solution of the given quadratic inequality, - x^2-4x+12≤0, consists of x-values for which the graph of the related quadratic function lies on and below the x-axis. The graph opens downward, since a= - 1 is less than zero.
We see that the graph lies on and below the x-axis at x≤ - 6 and x ≥ 2.
{ x| x ≤ - 6 or x ≥ 2 }
