Join IDNLearn.com to access a wealth of knowledge and get your questions answered by experts. Ask your questions and get detailed, reliable answers from our community of experienced experts.
Sagot :
Let's solve and simplify the given expressions step by step.
1. First Expression:
We need to multiply and simplify:
[tex]\[ (\sqrt{x} - 2\sqrt{2})(\sqrt{x} + 2\sqrt{2}) \][/tex]
This expression is in the form of a difference of squares:
[tex]\[ (a - b)(a + b) = a^2 - b^2 \][/tex]
where [tex]\( a = \sqrt{x} \)[/tex] and [tex]\( b = 2\sqrt{2} \)[/tex].
Applying the formula:
[tex]\[ (\sqrt{x} - 2\sqrt{2})(\sqrt{x} + 2\sqrt{2}) = (\sqrt{x})^2 - (2\sqrt{2})^2 \][/tex]
Simplify each term:
[tex]\[ (\sqrt{x})^2 = x \][/tex]
[tex]\[ (2\sqrt{2})^2 = 4 \cdot 2 = 8 \][/tex]
So the expression simplifies to:
[tex]\[ x - 8 \][/tex]
2. Second Expression:
Next, we need to simplify:
[tex]\[ (\sqrt{x} - \sqrt{2})^2 \][/tex]
This is a binomial squared, which can be expanded using the formula:
[tex]\[ (a - b)^2 = a^2 - 2ab + b^2 \][/tex]
where [tex]\( a = \sqrt{x} \)[/tex] and [tex]\( b = \sqrt{2} \)[/tex].
Applying the formula:
[tex]\[ (\sqrt{x} - \sqrt{2})^2 = (\sqrt{x})^2 - 2(\sqrt{x})(\sqrt{2}) + (\sqrt{2})^2 \][/tex]
Simplify each term:
[tex]\[ (\sqrt{x})^2 = x \][/tex]
[tex]\[ (\sqrt{2})^2 = 2 \][/tex]
[tex]\[ 2(\sqrt{x})(\sqrt{2}) = 2\sqrt{2x} \][/tex]
Combining these, we get:
[tex]\[ (\sqrt{x} - \sqrt{2})^2 = x - 2\sqrt{2x} + 2 \][/tex]
Therefore, the simplified forms of the expressions are:
[tex]\[ (\sqrt{x} - 2\sqrt{2})(\sqrt{x} + 2\sqrt{2}) = x - 8 \][/tex]
and
[tex]\[ (\sqrt{x} - \sqrt{2})^2 = x - 2\sqrt{2x} + 2 \][/tex]
Thus, the filled square in the final result box would be:
[tex]\[ (\sqrt{x}-\sqrt{2})^2 = x - 2\sqrt{2x} + 2 \][/tex]
1. First Expression:
We need to multiply and simplify:
[tex]\[ (\sqrt{x} - 2\sqrt{2})(\sqrt{x} + 2\sqrt{2}) \][/tex]
This expression is in the form of a difference of squares:
[tex]\[ (a - b)(a + b) = a^2 - b^2 \][/tex]
where [tex]\( a = \sqrt{x} \)[/tex] and [tex]\( b = 2\sqrt{2} \)[/tex].
Applying the formula:
[tex]\[ (\sqrt{x} - 2\sqrt{2})(\sqrt{x} + 2\sqrt{2}) = (\sqrt{x})^2 - (2\sqrt{2})^2 \][/tex]
Simplify each term:
[tex]\[ (\sqrt{x})^2 = x \][/tex]
[tex]\[ (2\sqrt{2})^2 = 4 \cdot 2 = 8 \][/tex]
So the expression simplifies to:
[tex]\[ x - 8 \][/tex]
2. Second Expression:
Next, we need to simplify:
[tex]\[ (\sqrt{x} - \sqrt{2})^2 \][/tex]
This is a binomial squared, which can be expanded using the formula:
[tex]\[ (a - b)^2 = a^2 - 2ab + b^2 \][/tex]
where [tex]\( a = \sqrt{x} \)[/tex] and [tex]\( b = \sqrt{2} \)[/tex].
Applying the formula:
[tex]\[ (\sqrt{x} - \sqrt{2})^2 = (\sqrt{x})^2 - 2(\sqrt{x})(\sqrt{2}) + (\sqrt{2})^2 \][/tex]
Simplify each term:
[tex]\[ (\sqrt{x})^2 = x \][/tex]
[tex]\[ (\sqrt{2})^2 = 2 \][/tex]
[tex]\[ 2(\sqrt{x})(\sqrt{2}) = 2\sqrt{2x} \][/tex]
Combining these, we get:
[tex]\[ (\sqrt{x} - \sqrt{2})^2 = x - 2\sqrt{2x} + 2 \][/tex]
Therefore, the simplified forms of the expressions are:
[tex]\[ (\sqrt{x} - 2\sqrt{2})(\sqrt{x} + 2\sqrt{2}) = x - 8 \][/tex]
and
[tex]\[ (\sqrt{x} - \sqrt{2})^2 = x - 2\sqrt{2x} + 2 \][/tex]
Thus, the filled square in the final result box would be:
[tex]\[ (\sqrt{x}-\sqrt{2})^2 = x - 2\sqrt{2x} + 2 \][/tex]
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com has the answers you need. Thank you for visiting, and we look forward to helping you again soon.