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

Sep 19, 2016

The sum of two squares cannot be factored.

#### Explanation:

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

$4 {x}^{2} + 25$ It cannot be factored with Real numbers

Only the difference of Two Squares can be factored.
$4 {x}^{2} - 25 = \left(x + 5\right) \left(x - 5\right)$

Sep 19, 2016

$4 {x}^{2} + 25 = \left(2 x - 5 i\right) \left(2 x + 5 i\right)$

#### Explanation:

The kind of factors or roots, a quadratic function $a {x}^{2} + b x + c$, depends critically on the discriminant ${b}^{2} - 4 a c$.

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 $4 {x}^{2} + 25$, discriminant is 0^2-4×4×25=0-400=-400. Hence factors / roots are complex.

We can factorize $4 {x}^{2} + 25$ as follows:

$4 {x}^{2} + 25$

= $4 {x}^{2} - \left(- 25\right)$

= ${\left(2 x\right)}^{2} - {\left(5 i\right)}^{2}$

= $\left(2 x - 5 i\right) \left(2 x + 5 i\right)$