Simplifying and solving for variables

[jannek.fahrenholz]

Hi

Simplifying and solving for variables

I am using guide two Manhattan GMAT, fifth edition. In chapter two and its problem set, precisely No 2 on page 31, there is the following equation:


x(x - ((5x+6)/x)) = 0

Solve for X!


With regard to the next page, 33, the solution shall be either 6 or -1. I do understand the solution but not its derivation.

Until step x^2-5x-6=0 everything is clear.

Then solution simplifies to (x-6) (x+1) = 0.

I do understand (x-6) (x+1) = x^2-5x-6 but is there an official way or even better a shortcut to see that x^2-5x-6 equals (x-6) (x+1)?

Thanks and Best Regs.

Jannek

[jannek.fahrenholz]

SOLVED ON MANGMAT (5th) GUIDE 2 PAGE 64

[jnelson0612]

Hi

Simplifying and solving for variables

I am using guide two Manhattan GMAT, fifth edition. In chapter two and its problem set, precisely No 2 on page 31, there is the following equation:


x(x - ((5x+6)/x)) = 0

Solve for X!


With regard to the next page, 33, the solution shall be either 6 or -1. I do understand the solution but not its derivation.

Until step x^2-5x-6=0 everything is clear.

Then solution simplifies to (x-6) (x+1) = 0.

I do understand (x-6) (x+1) = x^2-5x-6 but is there an official way or even better a shortcut to see that x^2-5x-6 equals (x-6) (x+1)?

Thanks and Best Regs.

Jannek

Hi Jannek,
Factoring quadratics is a skill that takes practice. Let's walk through the steps to factor x^2 - 5x - 6 to (x-6)(x+1).

First of all, let's look at how (x-6)(x+1) turns into x^2 - 5x - 6. That will help us when we break the quadratic down. Use FOIL: first, outside, inside, last.

(x+1)(x-6)
First: multiply the first terms of each of the parentheses together x * x = x^2
Outside: multiply the two outside terms together x * 1 = x
Inside: multiply the two inside terms together -6 * x = -6x
Last: multiply the two last terms together -6 * 1 = -6

Okay, so we have x^2, x, -6x, and -6. Add them together:
x^2 -5x - 6

So now let's look at how to break apart x^2 - 5x - 6 back into the two sets of parentheses:

1) Set up two sets of parentheses:
( )( )

2) Look at the first term in the quadratic, x^2. So I have an x * x. Put one x in each of the two parentheses as the first term:
(x )(x )

3) Look at the last term in the problem, -6. To get a negative number I multiply one positive number by one negative number. Add these signs to the two parentheses immediately following the x terms. We now have:
(x - )(x + )

4) This is the trickiest part. Think of the numbers that multiply together to make 6. I can think of 1 and 6 and 2 and 3. One will be negative and one will be positive.

Now, I look at the middle term. I have -5x. So the bigger number of either the pairs (1,6) or (2,3) must be negative and the smaller one is positive, since the negative ended up weighting more when I multiply each by x (as shown in the steps above). Let's consider:
-3x + 2x = -1x (doesn't match)
-6x + 1x = -5x (match!)

Thus, my last numbers must be a -6 and a 1.

I now can fully fill in my parentheses:
(x+1)(x-6) = x^2 - 5x -6.

I hope that this helps. This is a skill that you should practice if you are rusty because you will likely need this skill on the GMAT.

See if you can factor these:
x^2 + 7x + 10 = 0
x^2 - 4x + 3 = 0
x^2 - 2x - 8 = 0
Good luck! Let us know if we can help further.