Let's solve the equation [tex]\( x^2 = 20 \)[/tex] step-by-step to find the correct solutions.
1. Start with the given equation:
[tex]\[ x^2 = 20 \][/tex]
2. Take the square root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \pm \sqrt{20} \][/tex]
3. Simplify [tex]\( \sqrt{20} \)[/tex]:
- Break down 20 into its prime factors: [tex]\( 20 = 4 \times 5 \)[/tex].
- Therefore, [tex]\( \sqrt{20} = \sqrt{4 \times 5} \)[/tex].
- The square root of a product can be taken as the product of the square roots: [tex]\( \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} \)[/tex].
- We know that [tex]\( \sqrt{4} = 2 \)[/tex], so:
[tex]\[ \sqrt{20} = 2 \times \sqrt{5} \][/tex]
4. Include the [tex]\( \pm \)[/tex] to account for both positive and negative solutions:
[tex]\[ x = \pm 2 \sqrt{5} \][/tex]
So, the solutions to the equation [tex]\( x^2 = 20 \)[/tex] are [tex]\( x = 2 \sqrt{5} \)[/tex] and [tex]\( x = -2 \sqrt{5} \)[/tex].
Given the choices:
- A. [tex]\( x = \pm 5 \sqrt{2} \)[/tex]
- B. [tex]\( x = \pm 10 \sqrt{2} \)[/tex]
- C. [tex]\( x = \pm 2 \sqrt{10} \)[/tex]
- D. [tex]\( x = \pm 2 \sqrt{5} \)[/tex]
The correct answer is:
[tex]\[ \boxed{x = \pm 2 \sqrt{5}} \][/tex]
