Solve the inequality. x^3+9x^2-108 less than or equal to 0.
Then write in interval noation.
Please can you show me step by step on how you got the answer. I have tries and I am very confused.
first this software is very handy for polynomials and seeing their graphs:
http://mathportal.org/calculators/polynomials-solvers/polynomial-graphing-calculator.php
(x+6)(x+6)(x-3)
so we have double zero at x = -6 (graph comes up from way low and bounces back down off the x axis)
Then it comes back up again and goes positive at x = 3
SO
x^3+9x^2 -108 </= 0 for
x<+3
To solve the inequality x^3 + 9x^2 - 108 ≤ 0, you can follow these steps:
Step 1: Factor the expression, if possible.
In this case, the expression cannot be easily factored. So we move on to the next step.
Step 2: Determine the critical points.
The critical points occur when the expression is equal to zero. To find the critical points, set the expression x^3 + 9x^2 - 108 = 0 and solve for x.
(x - 6)(x^2 + 15x + 18) = 0
Setting each factor equal to zero:
x - 6 = 0 --> x = 6
x^2 + 15x + 18 = 0
Using the quadratic formula, we can solve for x:
x = (-15 ± √(15^2 - 4*1*18)) / (2*1)
x = (-15 ± √(225 - 72)) / 2
x = (-15 ± √153) / 2
Therefore, the critical points are x = 6 and x = (-15 ± √153) / 2.
Step 3: Test intervals and determine the sign of the expression.
To determine the sign of the expression in different intervals, we can test points within those intervals. We will test points within the intervals (-∞, (-15 - √153) / 2), ((-15 - √153) / 2, 6), (6, (-15 + √153) / 2), ((-15 + √153) / 2, +∞).
Let's test x = -16:
Substituting x = -16 into the expression x^3 + 9x^2 - 108:
(-16)^3 + 9(-16)^2 - 108 = -625
The expression is negative for x = -16.
Let's test x = -1:
Substituting x = -1 into the expression x^3 + 9x^2 - 108:
(-1)^3 + 9(-1)^2 - 108 = -100
The expression is also negative for x = -1.
Let's test x = 0:
Substituting x = 0 into the expression x^3 + 9x^2 - 108:
(0)^3 + 9(0)^2 - 108 = -108
The expression is negative for x = 0.
Let's test x = 7:
Substituting x = 7 into the expression x^3 + 9x^2 - 108:
(7)^3 + 9(7)^2 - 108 = 378
The expression is positive for x = 7.
Step 4: Write the solution in interval notation.
Based on the sign of the expression in the intervals we tested, we can determine the solution to be:
(-∞, (-15 - √153) / 2] ∪ (-1, 6] ∪ [(-15 + √153) / 2, +∞)
This is the solution in interval notation.
Please note that the solution could have also been obtained graphically by plotting the expression x^3 + 9x^2 - 108 on a graph and observing where it lies below the x-axis.