Are there any quadratic equations that cannot be solved by factoring? Why or why not?

Question

Are there any quadratic equations that cannot be solved by factoring? Why or why not?

Answer 1

Yes, only certain quadratic equations are factorable. All the rest that can't be factored? You'll have to resort to another method to solve those.

 

The a, b, and c coefficients of the quadratic equation must have a special relationship in order for it to be broken down into easy-to-handle factors. Here's a way to check:

 

- get the equation into ax² + bx + c = 0 form

- multiply a and c and get a product (this product is called ac)

- list all the factor pairs of ac

- if there's a factor pair that you can add or subtract together to equal the b coefficient, then the equation is factorable.

Answer 2

If the discriminant in the quadratic formula is a perfect square, then the quadratic expression can be factored.  

 

When we talk about factoring, the criteria we are talking about is to find four INTEGERS a, b, c and d such that the given quadratic expression can be obtained by expanding (ax+b)(cx+d)

 

If you take this expression as left side of quadratic equation as (ax+b)(cx+d)=0, we know that the solution set for this is {-b/a , -d/c}.  Both solutions are rational numbers and you can have rational solutions for a quadratic equation if (and only if) the discriminant is a perfect square