How do you simplify (x^2+2x-4)/(x^2+x-6)?

2 Answers
Jun 8, 2018

I don't think you can

Explanation:

We could simplify the fraction if the two polyonials shared a solution. In fact, let x_{n_1}, x_{n_2} be the roots of the numerator, and x_{d_1}, x_{d_2} be the roots of the denominator. This means that we could rewrite the fraction as

\frac{(x-x_{n_1})(x-x_{n_2})}{(x-x_{d_1})(x-x_{d_2})}

So, if x_{n_i}=x_{d_j} for some i,j=1,2, we could simplify that parenthesis.

Anyway, appling the quadratic formula, we have

x_{n_{1,2}} = \frac{-2\pm\sqrt(25)}{2} = -1\pm\sqrt{5}

and

x_{d_{1,2}} = \frac{-1\pm\sqrt(25)}{2} = \frac{-1\pm5}{2} = -3, 2

So, x_{n_1}, x_{n_2},x_{d_1} and x_{d_2} are all distinct, and we can't simplify anything.

Jun 8, 2018

(x^2+2x-4)/((x+3)(x-2))

Explanation:

Factorize first.

Step1: Factorize x^2+2x-4 by splitting the middle term.

Find two factors of -4 whose sum equals the coefficient of the middle term, which is 2

-4 + 1 = -3
-2 + 2 = 0
-1 + 4 = 3

Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Step 2: Factorize x^2+x-6 by splitting the middle term.

Find two factors of -6 whose sum equals the coefficient of the middle term, which is 1.

-6 + (-1) = 5
-3 +2 = -1
-2+3 = 1----> Correct!

x^2+x-6

x^2-2x+3x-6

x(x-2)+3(x-2)

(x+3)(x-2)

Hence the final simplification is:

(x^2+2x-4)/((x+3)(x-2))