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

2 Answers
Jun 29, 2017

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

Jun 29, 2017

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!