Express $\displaystyle \sqrt{44-24\sqrt{2}}$ in the form $\displaystyle a + b\sqrt{2}$
Hello, Mark!
$\displaystyle \text{We have: }\;\sqrt{44-24\sqrt{2}} \;=\;a + b\sqrt{2}\;\;\text{ where }a\text{ and }b\text{ are rational numbers.}$Express $\displaystyle \sqrt{44-24\sqrt{2}}$ in the form $\displaystyle a + b\sqrt{2}$
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: .$\displaystyle a + b\sqrt{2}\;=\;\begin{Bmatrix}6 - 2\sqrt{2} \\ \text{-}6 + 2\sqrt{2} \end{Bmatrix}$
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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