Solve the quadratic inequality x2-4x-21 ≤ 0 then write the solution in interval notation.

Answer 1

x² - 4x -21 ≤ 0

To find the solution to this inequality, we can break the quadratic into factors; (x+a)(x+b)

This is a case where the a and b will be integers; we just need to find the right combination:

-1, 21

1,-21

-3,7

3, -7

x² - 4x -21 = (x-7)(x+3)

so (x-7)(x+3) ≤ 0

Now we know the function on the left will equal zero at x = 7 and x = -3, so they will determine which regions will positive (the left side is qreater than 0) and negative (the left side is less than zero, which happens to be the region, we want to find)

if x is less than -3, both x-7 and x+3 will be less than zero and therefore negative, but their product will be positive and greater than zero.

if x >-3 and less than 7, x+3 will positive and x-7 will be negative and their product will be negative.

if x > 7. both x+3 and x-7 will be positive and their product will be positive,

so x² - 4x -21 ≤ 0 when -3 ≤ x ≤ 7 and using interval notation

the solution will be [-3,7] (notice that we use brackets instead of parentheses here because the solution INCLUDES -3 and 7)

Answer 2

Factor x²-4x-21 as (x-7)*(x+3)

determine where (x-7)*(x+3) = 0; x=7 and x=-3

Build a sign chart if you can.

You can also test different values. pick a number smaller than -3 say -10

for x=-10, (-10-7)*(-7+3) > 0 so x^2-4x-21 >= 0 on (-infinity, -3]

pick a number between -3 and 7 say 0

for x=0, (0-7)*(0+3) < 0 so x^2-4x-21 <= 0 on [-3, 7]

pick a number larger than 7 say 12

for x=12, (12-7)*(12+3) > 0 so x^2-4x-21 >= 0

determine the final answer