In this post, we will learn how to factorise the quadratic algebraic expression x^{2}+25, then solve the quadratic equation x^{2}+25=0.

**How to Factorise x**^{2}+25?

^{2}+25?

Answer: The factorization is given by x^{2}+25 = (x-5i)(x+5i), where i = √-1 is an imaginary complex number. |

**Solution:**

To factorise the expression x^{2}+25, we first write the expression in the form of a^{2}-b^{2}. This can be done as follows:

x^{2}+25

= x^{2} – (-25) as the negative of negative x is the number x itself.

= x^{2} – (-1 × 25)

= x^{2} – 25i^{2} where i = √-1

= x^{2} – (5i)^{2}

= (x-5i) (x+5i) using the formula a^{2}-b^{2} = (a-b)(a+b).

So the factors of x^{2}+25 is as follows: x^{2}+25 = (x-5i)(x+5i).

**How to Solve x**^{2}+25=0?

^{2}+25=0?

Answer: The solutions of x^{2}+25 = 0 are 5i, -5i where i = √-1 is an imaginary complex number. |

**Solution:**

**Method 1:** In the first method, we will solve the equation x^{2}+25 = 0 by factorization. The factorization of x^{2}+25 is given above.

x^{2}+25 = 0

⇒ (x-5i)(x+5i) = 0

So either x-5i=0 or x+5i =0

⇒ Either x=5i or x= -5i

So the solutions of x^{2}+25 = 0 are x=5i, -5i and thus there are two solutions of the equation x^{2}+25 = 0.

**Method 2:** Next, we will find the solutions of x^{2}+25 = 0 by the square root method.

x^{2}+25 = 0

⇒ x^{2} = -25

Taking square root on both sides, we get that

x = ± $\sqrt{-25}$ = ± $\sqrt{25 \times (-1)}$ = ± $\sqrt{25 \times i^2}$ as $i=\sqrt{-1}$.

⇒ x = ± 5i

⇒ x= 5i, -5i.

So 5i and -5i are the solutions of the given equation x^{2}+25=0.

**Video Solution of x ^{2}+25=0:**

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## FAQs

**Q1: How to factor x ^{2}+25?**

Answer: x^{2}+25 = (x-5i)(x+5i).

**Q2: Solve for x, x ^{2}+25=0.**

Answer: The solutions of x^{2}+25 = 0 are ± 5i.