How to factorize quadratic equations quickly?3n2−53n+232

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Celia Morse · Accepted Answer

Step 13n2−53n+232I use a process called &quot;A-C Method&quot;, &quot;Grouping&quot;, or &quot;Split the Middle Term&quot;.3⋅232=696Now, what two numbers add to -53 but multiply to 696?−23⋅−30? That&#039;s 690. Close.Now, keep in mind, (a+b)(a−b) is a maximum when b=0. Therefore, since we are at 690, and wish a higher result, the numbers need to get closer, so we just move by 1, since 690 was so close to 696.−24⋅−29? That&#039;s it! Second guess.3n2−24n−29n+232(we split the middle term so the 24 is a nice multiple of 3)(3n2−24n)−(29n−232)(Now, we group, and be mindful of that minus)3n(n−8)−29(n−8)(3n−29)(n−8)Will update/edit if OP needs clarification.

gonzakunti2 · Accepted Answer

Step 1As Raffaele says, it&#039;s tricky when the leading coefficient isn&#039;t 1, or a perfect square. Especially if you don&#039;t know if the expression is reducible or not. You can tell that by looking at the discriminant, b2−4ac, but if you&#039;ve gone that far, you might as well eliminate all the guesswork and just use the quadratic formula. ;)But anyway, assuming it is factorisable, because 3 is &#039;, −53&amp;lt;0 and 232&amp;gt;0, we know the factors must be of the form(3n−u)(n−v)with 3v+u=53 and uv=232  for u,v&amp;gt;0.Now 232=8⋅29 and any multiple of 29&amp;gt;53, which eliminates all possibilities except 3v+29=53. Thus u=29 and v=8, so the desired factorisation is(3n−29)(n−8)Step 2Another approach is to complete the square, but it is a bit tedious, and the numbers may get too large for rapid mental calculation.We need the leading coeffient to be a square, and the coefficient of the n term to be even. So we have to multiply this expression by 12. Thus36n2−12⋅53n+12⋅8⋅29(6n)2−2⋅6⋅53n+532−532+12⋅8⋅29(6n−53)2−(532−12⋅8⋅29)That constant term looks pretty bad, until we notice that53=29+24 and 4⋅24=12⋅8(6n−53)2−((29+24)2−4⋅24⋅29)(6n−53)2−(29−24)2(6n−53)2−52(6n−53+5)(6n−53−5)(6n−48)(6n−58)12(n−8)(3n−29)And now we can drop that multiplier of 12(n−8)(3n−29)As I said, it&#039;s a bit tedious, but we got there eventually.

How to factorize quadratic equations quickly? 3n^2 -53n + 232

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