Question

How do you factor `4x^2+25`?

Answer 1

The sum of two squares cannot be factored.

Explanation:

This expression can be described as the Sum of Two Squares.

`4x^2 +25` It cannot be factored with Real numbers

Only the difference of Two Squares can be factored.
`4x^2 -25 = (x+5)(x-5)`

Answer 2

`4x^2+25=(2x-5i)(2x+5i)`

Explanation:

The kind of factors or roots, a quadratic function `ax^2+bx+c`, depends critically on the discriminant `b^2-4ac`.

If the discriminant is `0`, we have equal roots. If it is square of a rational number, roots or factors are rational (provided `a` and `b` are rational). If discriminant is just positive, factors or roots are real and if discriminant is negative or a complex number, they are complex.

In the given polynomial `4x^2+25`, discriminant is `0^2-4×4×25=0-400=-400`. Hence factors / roots are complex.

We can factorize `4x^2+25` as follows:

`4x^2+25`

= `4x^2-(-25)`

= `(2x)^2-(5i)^2`

= `(2x-5i)(2x+5i)`