Factorise x^2 - 8x - 65?

x^2 - 8x - 65 =

(x - 13)(x + 5)

(x + 5)(x - 13). 13 and 5 have a product of 65, but to get a sum of -8, the 13 must be negative.

x^2 - 8x - 65

= x^2 - 13x + 5x - 65

= x(x - 13) + 5(x - 13)

= (x + 5)(x - 13)

Well 65 is 5 times 13, make one negative to get -65.

Which one? You need their sum to be -8 so better make 13 the negative one.

GET: (x-13)(x+5)

x^2 - 8x - 65

= x^2 - 13x + 5x - 65

= x(x - 13) + 5(x - 13)

= (x + 5)(x - 13)

A trick that is less used (because it is closer to art than science) is to split the middle term so that you create a pattern.

Here, because the constant (-65) is negative, the split has to involve a positive coefficient and a negative coefficient, adding up to -8.

Remember that you can write -8 as -9 + 1, or -10 + 2, or -11 + 3, etc.

Because the last term is divisible by 5 (and it is not divisible by 2 nor 3 -- therefore 5 is the first available divisor), do it in jumps of 5

-8 = -13 + 5, -18 + 10, -23 + 15, and so on.

Of course, in this case, things will work out on the first try (pure luck) and it looks like this

x^2 - 8x - 65

becomes

x^2 - 13x + 5x - 65

or

x^2 + 5x - 13x - 65

either one works.

In the first one, the pattern is +1::-13, +1::-13

Use parsing brackets (they change nothing but they make it easier to see what is going on):

(x^2 - 13x) + (5x - 65)

factor out the common factor in each bracket:

x(x - 13) + 5(x - 13)

Obviously, (x-13) is now a common factor for both terms, so we can factor it out

(x - 13)(x + 5)

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This method is based on the more familiar mathematical approach (finding two factors that add up to -8 and multiply out to -65), but is "easier" for people who prefer to manipulate terms instead of doing separate calculations. It is not better, it is not worst; it is simply different.

It is also a bit slower.

But I enjoy using it, because I feel I have some control on the terms.

(x - 13)(x + 5)

(x+a)(x+b) = x^2 + (a+b) + ab

So what two numbers sum to -8 (so at least one is negative) and multiply to give -65 (so only one is negative).

Ignoring ±65 and ∓1 (that sum to ±64), the only other factors of 65 are ±5 and ∓13, and if we want them to add to -8, then that is -13 and 5.

x^2 - 8x - 65 = (x+5)(x-13).

If it doesn't fall out that simply then you can factor by completing the square / completing the difference of two squares:

x² - 8x - 65

x² + 2(-4)x + (-4)² - (-4)² - 65

(x - 4)² - 81

(x - 4)² - 9²

(x - 4 - 9)(x - 4 + 9)

(x - 13)(x + 5)

(x - 13)*(x + 5)

If the constant term is negative, like -65 here, it means that the constants in the factored form (-13 and +5 here) have opposite signs. Does that help?

Do you know the abc-formula? Start by finding the roots

For the record, the answer is (x-13)(x+5)