Question

How do you factor the expression `x^2 + 17x + 52`?

Answer 1

`x^2+17x+52 = (x+13)(x+4)`

Explanation:

Note that `52 = 13xx4` and `17 = 13+4`

So: `x^2+17x+52 = (x+13)(x+4)`

In general we find `(x+a)(x+b) = x^2+(a+b)x+ab`

So if we can find a pair of numbers `a, b` whose product is the constant term and whose sum is the coefficient of the middle term, then we can factorise the quadratic as `(x+a)(x+b)`

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Alternative method

`x^2+17x+52` is in the form `ax^2+bx+c` with `a=1`, `b=17` and `c=52`.

This quadratic has zeros given by the quadratic formula:

`x = (-b+-sqrt(b^2-4ac))/(2a)`

`=(-17+-sqrt(17^2-(4*1*52)))/(2*1)`

`=(-17+-sqrt(289-208))/2`

`=(-17+-sqrt(81))/2`

`=(-17+-9)/2`

That is: `x = -13` or `x = -4`

Hence the quadratic has corresponding factors `(x+13)` and `(x+4)`