Question

Y=x^3+kx^2-9x-18

One solution to the equation shown above is -2. What is the positive solution to the equation?

Answer 1

3

Step-by-step explanation:

A solution to a function is when the graph of the function crosses the x-axis. In other words, it's a point where y = 0. So, we can set x^3 + kx^2 - 9x -18 equal to 0. Note that there is an unknown coefficient in this equation (k) that we have to solve for. We are given that one solution to this equation is -2. This means that we can plus x = -2 into the equation and solve for the unknown k:

`(-2)^3+k(-2)^2-9(-2)-18=0\\-8+4k+18-18=0\\4k=8\\k=2`

Therefore, the complete equation is the following:

`y=x^3+2x^2-9x-18`

As it turns out, we can use grouping to factor this polynomial. Let's start by re-ordering the terms to visualize it better:

`x^3-9x+2x^2-18=0`

The first two terms have a GCF of x, and the last two terms have a GCF of 2:

`x(x^2-9)+2(x^2-9)`

Now, there is a common term that we can bring to the front to complete our factoring process:

`(x^2-9)(x+2)`

First of all, note that x+2 is one of the factors, and if this was set to 0, we would get -2, which is consistent with the problem. Also, note that our other term can be factored more because it is a difference of two squares:

`(x-3)(x+3)(x+2)=0`

If we set each term to 0, we get x = 3, x = -3, and x = -2. We are only looking for the positive solution to this equation, which is 3