To express [tex]\((5 \sqrt{2} - \sqrt{5})(\sqrt{2} - \sqrt{5})\)[/tex] in the form [tex]\(a + b \sqrt{c}\)[/tex], we'll follow these steps:
1. Explicit Multiplication: First, distribute the terms using the distributive property (FOIL method).
[tex]\[
(5 \sqrt{2} - \sqrt{5})(\sqrt{2} - \sqrt{5}) = (5 \sqrt{2})(\sqrt{2}) + (5 \sqrt{2})(-\sqrt{5}) + (-\sqrt{5})(\sqrt{2}) + (-\sqrt{5})(-\sqrt{5})
\][/tex]
2. Simplify Each Product:
- [tex]\((5 \sqrt{2})(\sqrt{2})\)[/tex]: Multiplying the terms, we get [tex]\(5 \sqrt{2 \cdot 2} = 5 \sqrt{4} = 5 \cdot 2 = 10\)[/tex].
- [tex]\((5 \sqrt{2})(-\sqrt{5})\)[/tex]: Multiplying gives [tex]\(-5 \sqrt{2 \cdot 5} = -5 \sqrt{10}\)[/tex].
- [tex]\((-\sqrt{5})(\sqrt{2})\)[/tex]: Multiplying gives [tex]\(-\sqrt{5 \cdot 2} = -\sqrt{10}\)[/tex].
- [tex]\((-\sqrt{5})(-\sqrt{5})\)[/tex]: Multiplying gives [tex]\(\sqrt{5 \cdot 5} = \sqrt{25} = 5\)[/tex].
3. Combine Like Terms:
- From the first product, we get [tex]\(10\)[/tex].
- From the second and third products, we combine the terms with [tex]\(\sqrt{10}\)[/tex]:
[tex]\(-5 \sqrt{10} - \sqrt{10} = -6 \sqrt{10}\)[/tex].
- From the fourth product, we get [tex]\(5\)[/tex].
Putting it all together:
[tex]\[
10 + 5 - 6 \sqrt{10}
\][/tex]
4. Combine Constants:
- Combine the constant terms [tex]\(10 + 5\)[/tex]:
[tex]\[
15
\][/tex]
So, finally combining all terms, we have:
[tex]\[
15 - 6 \sqrt{10}
\][/tex]
The expression [tex]\((5 \sqrt{2} - \sqrt{5})(\sqrt{2} - \sqrt{5})\)[/tex] in the form [tex]\(a + b \sqrt{c}\)[/tex] is:
[tex]\[
a = 15, \quad b = -6, \quad \text{and} \quad c = 10
\][/tex]
Thus, the final result is:
[tex]\[
15 - 6 \sqrt{10}
\][/tex]
In this problem, we have:
[tex]\[
a = 0, \quad b = 8, \quad c = 2
\][/tex]
