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

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2 Answers

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.
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