Question

How do you solve `x^2+4x=285`?

Answer 1

15, - 19

Explanation:

Solving by the new Transforming Method (Google, Socratic Search).
`y = x^2 + 4x - 285 = 0`.
Find 2 real roots, with opposite signs (ac < 0), knowing sum (-b = -4) and product (-285).
Compose factor pairs of (-285) --> (5, - 57)(15, - 19). This last sum is
(15 - 19 = - 4 = - b).
The 2 real roots are: 15 and - 19.
Using AC Method.
Find 2 numbers, knowing sum (b = 4) and product (-285).
They are: (19, - 15) --> sum = 4 and product = -285
Split the term 4x into 19x and - 15 x
Factor by grouping:
x(x + 19) - 15(x + 19) = (x + 19)(x - 15) = 0
Solving binomials:
x + 19 = 0 --> x = - 19
x - 15 = 0 --> x = 15

Answer 2

Here's a simpler way to solve: by completing the square.
(Also the answer is `x = -15` and `x = 19`)

Explanation:

To solve by completing the square we first have to realize that `x^2 +4x = 285` is the same as `x^2 +4x - 285 = 0`. Look familiar? That's because it is.

1.) The first step in completing the square is to move `c` to the other side of the equation (which was already done for us above).

2.) Now that we have the equation `x^2 + 4x = 285`, we are going to try and turn the left side of the equation into a perfect square trinomial so that we can factor it (why we are doing this will be obvious later). In order to turn the left side of the equation into a perfect square trinomial, we are going to divide `b` by `2`, square the result, and add it to both sides.

`4/2 = (2) ^2 = 4` Now add `4` to both sides to make sure the equation stays balanced.

`x^2 + 4x + 4 = 289`

3.) Does anything about the left side of the equation catch your eye? It should because the left side is now a perfect square trinomial, and can be factored, and when you factor it, you are left with the equation `(x+2)^2 = 289`. Now we can finally solve for `x`!

4.) Take the square root of both sides of the equation to isolate `(x+2)` and you are left with `x+2 = +-17`

5.) Because there is a `+-17`, you have to solve for `x` twice:

`x-2 = 17`
`x-2 = -17`

6.) By solving for `x`, you now see that `x = 19` & `x= -15`

Hope this helps!