Question

Solve `2x^3 + x^2 = -4 - 8x`?

Answer 1

`x = -1/2`

graph{2x^3 + x^2 + 8x + 4 [-11.06, 11.44, -4.63, 7.09]}

Explanation:

First thing you always want to do when solving polynomial equations is set them equal to zero. So:

`2x^3 + x^2 = -4 - 8x`
`=> 2x^3 + x^2 + 8x + 4 = 0`

Now, we're going to use a method of solving called grouping. We're going to split the left hand side of our equation into two groups of 2 terms each, and then try to factor out some common term out of each group.

`=> (2x^3 + x^2) + (8x + 4) = 0`

I see that I can factor out a `2x + 1` out of each of my groups. This would leave:

`=> (2x + 1)(x^2) + (2x+1)(4) = 0`

Since I have a `2x + 1` in each of my terms, I can factor it out, and clump what's left together:

`=> (2x+1)(x^2 + 4) = 0`

Now that I have a product of factors, I can invoke my zero product property, and know that for this equation to be true, one of those factors must equal zero.

`=> 2x + 1 = 0`
`x = -1/2`

`=> x^2 + 4 = 0`
`x = +-sqrt(-2)`

...but wait, how can we have a negative number under our square root? The answer is we cannot! That is, we cannot have a negative number inside a square root and expect a real number as an answer. So your only real solution to this equation would be `x = -1/2`. However, if we were to consider imaginary solutions, we'd also include:

`x = +-isqrt(2)`

However, you should only ever include this in your answer if it imaginary solutions are specifically asked for.

A handy way to check your answer right after is to graph it. Let's see how that turns out:

graph{2x^3 + x^2 + 8x + 4 [-11.06, 11.44, -4.63, 7.09]}

You'll see that our graph does in fact intersect the x-axis at `x = -1/2`, meaning that we are correct.

Here's a great video by patrickJMT if you want to learn more about the process of grouping;

enter link description here

Hope that helps :)