# How to factor the equation x^2+13=0.

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### 3 Answers

We have to factor the equation x^2 + 13 = 0

Now we can write x^2 + 13 = 0 in the form (x - a)(x - b) =0

where a and b would be the roots of the equation.

For x^2 + 13 = 0

x^2 = -13

=> x = -i*sqrt 13 and +i*sqrt 13

a = -i*sqrt 13 and b = +i*sqrt 13

So (x - a)(x - b) =0

=> ( x + i*sqrt 13)(x - i*sqrt 13) = 0

**The required result is x^2 + 13 = ( x + i*sqrt 13)(x - i*sqrt 13) = 0**

x^2 + 13 = 0

We will factor using the difference between the squares.

We know that:

a^2 -b^2 = (a-b)(a+b)

let us rewrite:

x^2 - (-13) = 0

Now we will factor:

==> (x^2-(-13) = 0

==> (x-sqrt-13) ( x+sqrt-13) = 0

But we know that sqrt-13 = sqrt13*i

==> (x-sqrt13*i)(x+sqrt13*i) = 0

==> x^2 +13 = (x-sqrt13*i)(x+sqrt13*i) = 0

**==> x1= sqrt13*i**

**==> x2= -sqrt13*i**

x^2 + 13 = 0

We'll subtract 13:

x^2 = -13

x1 = sqrt(-13)

We'll recall the fact that sqrt -1 = i (imaginary unit):

We'll re-write the result:

x1 = i*sqrt 13

x2 = - i*sqrt 13

Since we have found out the roots of the quadratic, we'll factorize the equation:

x^2 + 13 = (x - x1)(x - x2)

**x^2 + 13 = (x - i*sqrt 13)(x + i*sqrt 13)**