When to use factoring by grouping for quadratic equation

Question

There are several ways/methods to perform factoring. I am revising factoring at KhanAcademy, there are factoring by grouping, factoring special product and factoring difference of squares.

Although, I can work on the exercise, but I do not really understand when to apply factoring by grouping.

Answer 1

Suppose your quadratic factors as $$(ax+b)(cx+d)$$ Then your quadratic was $$acx^2+(ad+bc)x+bd$$ This can be factored by grouping if you can hit upon the correct way of splitting the coefficient of $x$ into two pieces, $$acx^2+(ad+bc)x+bd=(acx^2+adx)+(bcx+bd)=ax(cx+d)+b(cx+d)=(ax+b)(cx+d)$$ In other words, any quadratic that factors can, in principle, be factored by grouping, but you have to be clever/lucky to find the right grouping. E.g., given $$210x^2+421x+210$$ there are a lot of way to split $421$ into a sum of two numbers; how do you know that the way that will work is $421=225+196$? $$210x^2+421x+210=210x^2+225x+196x+210=15x(14x+15)+14(14x+15)=(15x+14)(14x+15)$$

Answer 2

If a quadratic has the form $$ax^2+bx+c$$ then I would look at the discriminant of the quadratic formula (i.e. $b^2-4ac$) to see if it is a perfect square. If it is, then I would try to factor by grouping (or difference of squares if $b=0$). If not, then I would use the quadratic formula. By following these steps, your teacher will receive positive feedback that the brainwashing is, in fact, working.

In the real world though, we ALWAYS use the quadratic formula. Factoring by grouping is almost always completely useless in any kind of an experimental or scientific scenario. We would be very lucky, indeed, to encounter coefficients of a quadratic that would allow a grouping solution. In class we are given contrived problems like

A garden that is 4 meters wide and 6 meters long is to have a uniform border such that the area of the border is the same as the area of the garden. Find the width of the border.

to make it seem to student as though grouping may have some practical application.

If grouping is so useless, then why bother teaching it? Is it a union conspiracy to maximize the number of employed teachers? I think not. The reason why we teach factoring by grouping is to give students at least some exposure in high school to Diophantine Equations. As a continuation of Least Common Multiple, Greatest Common Divisor, Prime Factorization, and sum of digits analysis, it represents the final topic in Number Theory that students will be exposed to prior to University. As such, it is an important part of their education. I would have appreciated it, though, had my math teacher been more upfront about it.

Answer 3

Short answer: factoring by grouping IS factoring. All those other factoring techniques are just special cases of factoring by grouping, and once you understand how to do factoring by grouping you automatically can derive all of the special cases.

In general if you want to find the roots of a quadratic equation, this is the approach you should use:

1. Look for special factorizations of the form $x^2 \pm 2a + a^2$ or $x^2 - a^2$. If you find one of these, you're already done.
2. If the equation looks simple (or your homework assignment says you have to), use factoring by grouping.
3. Otherwise, use the quadratic formula.

What does it mean to have a "simple" equation? Basically, a quadratic equation $ax^2 + bx + c$ where the numbers $a$,$b$,$c$ are integers, or in trickier cases, simpler fractions or even radicals. $x^2 + 5x + 6$, $x^2 + \frac{3}{4}x + \frac{1}{8}$ and $x^2 + 2\sqrt{3}x + 3$ would qualify. (Actually, that last one is a special factorization, $(x+\sqrt{3})^2$.) Once you get more experience, you'll be able to see at once whether an equation is "simple" and thus worth factoring.

Some quadratics look easy but their roots are near impossible to find without the quadratic formula, for instance $x^2 + 5x + 7$. With other equations as well as equations with long decimals like $1.618x^2 + 5.1234232x + 4.329$ (the stuff that comes up more in the "real world"), it's not even worth trying. Just use the quadratic formula.

Factoring by grouping isn't restricted to quadratic polynomials. Although it becomes much more difficult, you can factor polynomials of any degree using this method.