Find the solution(s) that satisfy both equations:
a) x^3 + x^2 -20x <= 20 ||| b) x^3 - 3x^2 - 4x + 12 > 0
It might be easier to see the solution set when graphing them if you graph
x^3 + x^2 -20x - 20 <= 0
x^3 - 3x^2 - 4x + 12 > 0
That way the intervals are bounded by the roots of the functions. Come back if you get stuck. If you are having trouble calculating the roots, note that both can be factored easily by grouping.
So I factored by grouping and got
a) x[(x+5)(x-4)] <= 0
b) (x-2)(x+2)(x-3) > 0
I plugged the numbers into desmos, but I'm still having difficulty understanding what exactly this means
View the graphs here:
www.wolframalpha.com/input/?i=plot+x%5E3+%2B+x%5E2+-20x+-+20%2C+x%5E3+-+3x%5E2+-+4x+%2B+12
You can see that both conditions apply only on the interval
[-1,2)U[3,2√5)
Your factoring is also off
x^3 + x^2 -20x - 20 = (x+1)(x-√20)(x+√20)
Thank you oobleck, just one more thing:
I think you accidentially put the wrong equation, my equation is x^3 + x^2 -20x <= 0, *NOT* x^3 + x^2 -20x - 20.
Actually, on closer analysis, I also got my on equation wrong, I put x^3 + x^2 -20x <= 20 when in reality it is x^3 + x^2 -20x <= 0
To find the solution(s) that satisfy both equations, we need to solve each equation separately and then find the intersection of their solutions.
a) To solve the equation x^3 + x^2 - 20x <= 20, we can follow these steps:
Step 1: Rewrite the inequality as an equation:
x^3 + x^2 - 20x = 20
Step 2: Move all terms to one side of the equation:
x^3 + x^2 - 20x - 20 = 0
Step 3: Factor the equation if possible:
The given equation does not factor easily, so we can use numerical methods to approximate the solutions.
Step 4: Use a numerical method (such as graphing or using a calculator) to find the solutions:
Using a graphing calculator or software, plot the function f(x) = x^3 + x^2 - 20x - 20 and find the x-values where the graph is equal to zero. These x-values will approximate the solutions to the equation.
b) To solve the equation x^3 - 3x^2 - 4x + 12 > 0, we can follow these steps:
Step 1: Rewrite the inequality as an equation:
x^3 - 3x^2 - 4x + 12 = 0
Step 2: Move all terms to one side of the equation:
x^3 - 3x^2 - 4x + 12 = 0
Step 3: Factor the equation if possible:
The given equation does not factor easily, so we can use numerical methods to approximate the solutions.
Step 4: Use a numerical method (such as graphing or using a calculator) to find the solutions:
Using a graphing calculator or software, plot the function f(x) = x^3 - 3x^2 - 4x + 12 and find the x-values where the graph is greater than zero. These x-values will approximate the solutions to the inequality.
Finally, find the intersection of the solutions from both equations to find the solution(s) that satisfy both equations.