olve the following quadratic inequality by determining the roots, sketch the parabola in relation to the x axis and determine if the parabola is above or below

Question

solve the following quadratic inequality by determining the roots, sketch the parabola in relation to the x axis and determine if the parabola is above or below the x axis  

 

-x^2-x+2 greater than or equal to 0

Answer 1

-x² - x + 2 ≥ 0

 

(-x + 1)(x + 2) ≥ 0

 

Set the factors greater or equal to zero.

 

 

-x + 1 ≥ 0          and          x + 2 ≥ 0

 

-x ≥ -1               and          x ≥ -2

 

x ≥ 1

 

x ≤ 1

 

Next, we perform test points.  We will use x=-3 , x=0 , and x=2 and plug them in to the factored form of quadratic, since -2 and 1 are our zeros.

 

f(-3) = (-(-3) + 1)(-3 + 2)

        = (3 + 1)(-1)

        = 4(-1)

        = -4

 

f(0) = (0 + 1)(0 + 2)

      = 1(2)

      = 2

 

f(2) = (-2 + 1)(2 + 2)

      = (-1)(4)

      = -4

 

At x=0, f(x) is positive.  This means the f(x) is above the x-axis.  At x=-3 and x=2, f(x) is negative.  This means that f(x) is below the x-axis.

 

 

Intervals where f(x) is above the x-axis:

 

(-2, 1)

 

 

Intervals where f(x) is below the x-axis:

 

(-∞, -2)∪(1, ∞)

 

 

Since the leading term of the equation is negative, the parabola will open downward.