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(x^2+4) Factored

2 min read Jun 17, 2024
(x^2+4) Factored

Factoring (x^2 + 4)

The expression (x^2 + 4) cannot be factored using real numbers. Here's why:

Understanding Factoring

Factoring involves breaking down an expression into its simpler multiplicative components. We typically look for two binomials that multiply together to give the original expression.

The Problem with (x^2 + 4)

  • No Real Roots: The quadratic equation x^2 + 4 = 0 has no real solutions. This means there are no real numbers that, when squared, result in -4.
  • Difference of Squares: Factoring often relies on the difference of squares pattern: (a^2 - b^2) = (a + b)(a - b). However, (x^2 + 4) is a sum of squares, not a difference.

Factoring with Complex Numbers

While it can't be factored using real numbers, (x^2 + 4) can be factored using complex numbers. Complex numbers involve the imaginary unit "i," where i^2 = -1.

Here's how it works:

  1. Rewrite the expression: x^2 + 4 = x^2 - (-4)
  2. Introduce the imaginary unit: x^2 - (-4) = x^2 - (2i)^2
  3. Apply the difference of squares pattern: x^2 - (2i)^2 = (x + 2i)(x - 2i)

Therefore, the factored form of (x^2 + 4) using complex numbers is (x + 2i)(x - 2i).

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