[SOLVED] Expressing in the form?

• Jun 12th 2009, 06:53 PM
mark1950
[SOLVED] Expressing in the form?
Express $\displaystyle \sqrt{44-24\sqrt{2}}$ in the form $\displaystyle a + b\sqrt{2}$
• Jun 12th 2009, 07:40 PM
Soroban
Hello, Mark!

Quote:

Express $\displaystyle \sqrt{44-24\sqrt{2}}$ in the form $\displaystyle a + b\sqrt{2}$
$\displaystyle \text{We have: }\;\sqrt{44-24\sqrt{2}} \;=\;a + b\sqrt{2}\;\;\text{ where }a\text{ and }b\text{ are rational numbers.}$

Square both sides: .$\displaystyle 44 - 24\sqrt{2} \:=\:(a+b\sqrt{2})^2 \:=\:a^2 + 2ab\sqrt{2} + 2b^2$

And we have: .$\displaystyle (a^2 + 2b^2) + (2ab)\sqrt{2} \;=\;44-24\sqrt{2}$

Equate coefficients: .$\displaystyle \begin{array}{ccc}a^2+b^2 \:=\:44 & {\color{blue}[1]} \\ 2ab \:=\:-24 & {\color{blue}[2]} \end{array}$

From $\displaystyle {\color{blue}[2]}\!:\;\;b \:=\:-\frac{12}{a}\;\;{\color{blue}[3]}$

$\displaystyle \text{Substitute into }{\color{blue}[1]}\!:\;\;a^2 + 2\left(-\frac{12}{a}\right)^2 \:=\:44 \quad\Rightarrow\quad a^2 + \frac{288}{a^2} \:=\:44$

Multiply by $\displaystyle a^2\!:\;\;a^4 + 288 \:=\:44a^2 \quad\Rightarrow\quad a^4 - 44a^2 + 288 \:-\:0$

Factor: .$\displaystyle (a^2 - 36)(a^2-8) \:=\:0 \quad\Rightarrow\quad (a-6)(a+6)(a^2-8) \:=\:0$

. . And the rational roots are: .$\displaystyle a \:=\:\pm 6$

Substitute into $\displaystyle {\color{blue}[3]}\!:\;\;b \:=\:-\frac{12}{\pm6} \:=\:\mp 2$

There are two solutions: .$\displaystyle a + b\sqrt{2}\;=\;\begin{Bmatrix}6 - 2\sqrt{2} \\ \text{-}6 + 2\sqrt{2} \end{Bmatrix}$

• Jun 13th 2009, 02:08 AM
mark1950
What's the difference between $\displaystyle \pm$ and $\displaystyle \mp$? Why do you have to inverse them to get $\displaystyle \mp2$? Isn't it okay to just put $\displaystyle \pm2$?
• Jun 13th 2009, 02:13 AM
yeongil
Quote:

Originally Posted by mark1950
What's the difference between $\displaystyle \pm$ and $\displaystyle \mp$? Why do you have to inverse them to get $\displaystyle \mp2$? Isn't it okay to just put $\displaystyle \pm2$?

No. Looking at this from Soroban's post:
$\displaystyle b \:=\:-\frac{12}{\pm6} \:=\:\mp 2$

See the $\displaystyle \pm6$ in the denominator? The $\displaystyle \mp$ sign indicates that the number that follows has to be the opposite of whatever sign the number after the $\displaystyle \pm$ is. If the denominator is +6, then it simplifies to -2, and if the denominator is -6, it simplifies to +2.

The $\displaystyle \mp$ isn't used as often as $\displaystyle \pm$. If you know trig, then you may have seen the sum/difference identity for cosine combined into one identity like this:
$\displaystyle \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$.

Of course, the two identities are these:
$\displaystyle \cos(A + B) = \cos A \cos B - \sin A \sin B$.
$\displaystyle \cos(A - B) = \cos A \cos B + \sin A \sin B$.

Note how the sign changes on the right side. That's the reason we use the $\displaystyle \mp$ sign.

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