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Intern  B
Joined: 15 Oct 2017
Posts: 28
Schools: Northeastern '20
Linear Equation question  [#permalink]

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Hello,

Steep 1 : x²-5x-6=0

Steep 2 : (x-6)(x+1)

How should we make the transition? I don't get it

Thanks!
Senior PS Moderator D
Status: It always seems impossible until it's done.
Joined: 16 Sep 2016
Posts: 737
GMAT 1: 740 Q50 V40 GMAT 2: 770 Q51 V42 Re: Linear Equation question  [#permalink]

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thegame12 wrote:
Hello,

Steep 1 : x²-5x-6=0

Steep 2 : (x-6)(x+1)

How should we make the transition? I don't get it

Thanks!

Hi thegame12,

Step 2 is just factoring the given quadratic expression from step 1.

You can think like this...
(x - a)*(x - b) = x*x + x*(-b) + x*(-a) + a*b
= x^2 - (a+b)x +ab

You can remember the above as a result. Linear term is always sum of roots and constant term is the product.

So for given expression a+b=5 & a*b =-6
This is true for a = 6 and b = -1. ( You can tell this by trying out small numbers )

So step 2: (x - 6)*(x - (-1)) = (x-6)*(x+1)

Does that make sense?

Best,
_________________
Regards,

“Do. Or do not. There is no try.” - Yoda (The Empire Strikes Back)
Math Expert V
Joined: 02 Sep 2009
Posts: 58320
Re: Linear Equation question  [#permalink]

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Manhattan Prep Instructor G
Joined: 04 Dec 2015
Posts: 833
GMAT 1: 790 Q51 V49 GRE 1: Q170 V170 Re: Linear Equation question  [#permalink]

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thegame12 wrote:
Hello,

Steep 1 : x²-5x-6=0

Steep 2 : (x-6)(x+1)

How should we make the transition? I don't get it

Thanks!

You want to find two numbers - let's call them A and B - that have the following properties:

- when you multiply them together, you get the last number in the quadratic (in this case, -6. Be careful about the negative sign!)
- when you add them together, you get the middle number in the quadratic (in this case, -5.)

So, what two numbers add together to make -5, and multiply together to make -6?

The easiest place to start is to think about which numbers multiply to -6, then try adding them together and see what happens.

For instance, -3*2 = -6, but -3+2 doesn't equal -5. So that isn't the right pair of numbers.

But, -6*1 = -6, and -6+1 = -5. That's the right pair of numbers.

Once you have those numbers, you can write the quadratic like this:

(x-6)(x+1) = 0

Again, be careful to keep the negative signs the same as what you figured out earlier!

Finally, two things to be careful about:
- you can only use this technique when the first term in the quadratic doesn't have a coefficient. In other words, you can only do this when the quadratic starts with just $$x^2$$ (or any variable squared), not when it starts with something like $$2x^2$$. If it does, you have to divide the whole quadratic to make the 2 go away first.
- this technique is only useful when the quadratic is equal to 0. That is, when there's a 0 on the other side of the equals sign. The whole point of doing this is to find the values of x, and it's only easy to do that when you know that the product comes out to 0.
_________________ Chelsey Cooley | Manhattan Prep | Seattle and Online

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Manager  G
Joined: 16 Oct 2011
Posts: 106
GMAT 1: 570 Q39 V41 GMAT 2: 640 Q38 V31 GMAT 3: 650 Q42 V38 GMAT 4: 650 Q44 V36 GMAT 5: 570 Q31 V38 GPA: 3.75
Re: Linear Equation question  [#permalink]

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if a*b = 0 then (1) a=0 (2) b=0, or both a=0 and b=0. We can set each binomial factor as a linear equation. x-6=0-->x=6, x+1=0--->x=-1 Re: Linear Equation question   [#permalink] 10 Apr 2018, 08:14
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