Answered

In this question, you must show detailed reasoning. Solutions relying on calculator technology are not acceptable.

(a) Write [tex]\frac{6}{\sqrt{5}-\sqrt{2}}[/tex] in the form [tex]a\sqrt{5} + b\sqrt{2}[/tex], where [tex]a[/tex] and [tex]b[/tex] are integers to be found.

(b) Hence, or otherwise, solve the equation [tex]\sqrt{5}x = \sqrt{2}x + 18\sqrt{5}[/tex], giving your answer in its simplest form.



Answer :

Sure, let's work through each part of the question step-by-step.

### Part (a)

We are asked to write [tex]\(\frac{6}{\sqrt{5} - \sqrt{2}}\)[/tex] in the form [tex]\(a \sqrt{5} + b \sqrt{2}\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers.

First, we need to rationalize the denominator. To do this, we multiply the numerator and the denominator by the conjugate of the denominator, which is [tex]\(\sqrt{5} + \sqrt{2}\)[/tex]:

[tex]\[ \frac{6}{\sqrt{5} - \sqrt{2}} = \frac{6 (\sqrt{5} + \sqrt{2})}{(\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2})} \][/tex]

Now, let's simplify the denominator:

[tex]\[ (\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2}) = \sqrt{5}^2 - \sqrt{2}^2 = 5 - 2 = 3 \][/tex]

So the expression becomes:

[tex]\[ \frac{6 (\sqrt{5} + \sqrt{2})}{3} \][/tex]

Now, simplify the numerator:

[tex]\[ \frac{6 (\sqrt{5} + \sqrt{2})}{3} = 2 (\sqrt{5} + \sqrt{2}) = 2 \sqrt{5} + 2 \sqrt{2} \][/tex]

Thus, [tex]\(\frac{6}{\sqrt{5} - \sqrt{2}}\)[/tex] can be written in the form [tex]\(a \sqrt{5} + b \sqrt{2}\)[/tex] where [tex]\(a = 2\)[/tex] and [tex]\(b = 2\)[/tex].

### Part (b)

Now, we need to solve the equation

[tex]\[ \sqrt{5} x = \sqrt{2} x + 18 \sqrt{5} \][/tex]

First, let's isolate the terms involving [tex]\(x\)[/tex]. Subtract [tex]\(\sqrt{2} x\)[/tex] from both sides of the equation:

[tex]\[ \sqrt{5} x - \sqrt{2} x = 18 \sqrt{5} \][/tex]

Factor out [tex]\(x\)[/tex] on the left side:

[tex]\[ x (\sqrt{5} - \sqrt{2}) = 18 \sqrt{5} \][/tex]

To solve for [tex]\(x\)[/tex], divide both sides by [tex]\(\sqrt{5} - \sqrt{2}\)[/tex]:

[tex]\[ x = \frac{18 \sqrt{5}}{\sqrt{5} - \sqrt{2}} \][/tex]

We already rationalized a similar expression in part (a), where we found that:

[tex]\[ \frac{6}{\sqrt{5} - \sqrt{2}} = 2 \sqrt{5} + 2 \sqrt{2} \][/tex]

So, we can adjust this result to find the equivalent form for our current expression. Notice:

[tex]\[ \frac{18 \sqrt{5}}{\sqrt{5} - \sqrt{2}} = 3 \cdot \frac{6 \sqrt{5}}{\sqrt{5} - \sqrt{2}} = 3 \left( 2 \sqrt{5} + 2 \sqrt{2} \right) \][/tex]

Thus, we get:

[tex]\[ x = 3 \left( 2 \sqrt{5} + 2 \sqrt{2} \right) \][/tex]

Simplifying this:

[tex]\[ x = 6 \sqrt{5} + 6 \sqrt{2} \][/tex]

Therefore, the solution to the equation [tex]\(\sqrt{5} x = \sqrt{2} x + 18 \sqrt{5}\)[/tex] is:

[tex]\[ x = 6 \sqrt{5} + 6 \sqrt{2} \][/tex]

In its simplest form.