# How did they get x^2 - 4x + 3 = 0 => (x-3)(x-1)? Do you do it by intuition or is there something to memorize?

How did they get (x-3)(x-1)

How did they factor it? Do you just do it by intuition or is there something to memorize?

### 3 Answers

- PuzzlingLv 71 month agoFavourite answer
x² - 4x + 3 = 0

Usually they'll give you problems that are easy enough to figure out with integer factors.

You are looking for two numbers:

They should *multiply* to be the last number (3)

They should *add* to be the middle coefficient (-4)

Clearly 3 is positive. You either need two positive numbers or two negative numbers to get a positive product. The only possible products are:

1 * 3 = 3

-1 * -3 = 3

But obviously since the middle coefficient (-4) is negative, we want the second pair. You can't get a negative sum from two positive numbers.

-1 + -3 = -4

From there, you just write it as:

(x - 1)(x - 3) = 0

Later on you'll learn more methods for doing this that don't rely on factoring such as completing the square or the quadratic formula.

Summary:

Think of two numbers that multiply to be the last number.

- You'll need two numbers of the same sign if the number is positive. You'll need two numbers of opposite signs if the number is negative.

- Those same numbers need to *add* to be the middle coefficient.

- KrishnamurthyLv 71 month ago
x^2 - 4x + 3 = 0

x^2 - x - 3x + 3 = 0

x(x - 1) - 3(x - 1) = 0

(x - 1)(x - 3) = 0

- AlanLv 71 month ago
For quadratic equation,

there is always

the quadratic formula

(-b +/- sqrt( b^2 -4ac) )/ 2a

where form is ax^2 + bx +c = 0

so a = 1 , b = -4 , c= 3 then

plug into formula

(+4 +/- sqrt( 16 -12) ) / 2 = 2 +/- sqrt(4)/2

= 2 +/- 1 = 1 or 3

root of 1 and 3

means factor

(x-1)(x-3)

otherwise, you a few things to help

1st rational roots theorem

so all possible roots of al

are all factor of p/q where p is the lowest coefficient

and q the highest

+/- 3 , +/- 1

ax^2 + bx + c = (x+ f)(x+g) = x^2 + (f+g)x + fg

b = (f+g)

c = fg

so you have try the possibilites

or +/- 3, +/- 1 which

-4 = f + g

and

3 = fg

just check the values.

There is a 3rd method using

a completing of squares.