Question
If $\text{a}=3-2\sqrt{2},$ find the value of $\text{a}^2-\frac{1}{\text{a}^2}.$
✓
Answer
$\text{a}=3-2\sqrt{2}$
$\Rightarrow\text{a}^2=\big(3-2\sqrt{2}\big)^2$
$=3^2-2\times3\times2\sqrt{2}+\big(2\sqrt{2}\big)^2$
$=9-12\sqrt{2}+8$
$=17-12\sqrt{2}$
$\Rightarrow\frac{1}{\text{a}^2}=\frac{1}{17-12\sqrt{2}}$
$=\frac{1}{17-12\sqrt{2}}\times\frac{17+12\sqrt{2}}{17+12\sqrt{2}}$
$=\frac{17+12\sqrt{2}}{17^2-\big(12\sqrt{2}\big)^2}$
$=\frac{17+12\sqrt{2}}{289-288}$
$=17+12\sqrt{2}$
$\Rightarrow\text{a}^2-\frac{1}{\text{a}^2}=\big(17-12\sqrt{2}\big)-\big(17+12\sqrt{2}\big)$
$=17-12\sqrt{2}-17-12\sqrt{2}$
$=-24\sqrt{2}$
$\Rightarrow\text{a}^2=\big(3-2\sqrt{2}\big)^2$
$=3^2-2\times3\times2\sqrt{2}+\big(2\sqrt{2}\big)^2$
$=9-12\sqrt{2}+8$
$=17-12\sqrt{2}$
$\Rightarrow\frac{1}{\text{a}^2}=\frac{1}{17-12\sqrt{2}}$
$=\frac{1}{17-12\sqrt{2}}\times\frac{17+12\sqrt{2}}{17+12\sqrt{2}}$
$=\frac{17+12\sqrt{2}}{17^2-\big(12\sqrt{2}\big)^2}$
$=\frac{17+12\sqrt{2}}{289-288}$
$=17+12\sqrt{2}$
$\Rightarrow\text{a}^2-\frac{1}{\text{a}^2}=\big(17-12\sqrt{2}\big)-\big(17+12\sqrt{2}\big)$
$=17-12\sqrt{2}-17-12\sqrt{2}$
$=-24\sqrt{2}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
If both $x+1$ and $x-1$ are factors of $a x^3+x^2-2 x+b$, find the values of $a$ and $b$.
→How many cubic meters of earth must be dug out to sink a well $21\ m$ deep and 6m diameter? Find the cost of plastering the inner surface as well at ₹ $9.50$ per $m^2.$
→Prove that $(a + b + c)^3 - a^3 - b^3 - c^3 = 3(a + b )(b + c)(c + a).$
→In the given figure, $ABCD$ is a quadrilateral inscribed in a circle with centre $O$. $CD$ is produced to $E$ such that $\angle\text{AED} = 95^\circ$ and $\angle\text{OBA} = 30^\circ$ Find $\angle\text{OAC.}$
→ABC is a triangle and through A, B, C lines are drawn parallel to BC, CA and AB respectively intersecting at P, Q and R. Prove that the perimeter of $\triangle\text{PQR}$ is double the perimeter of $\triangle\text{ABC}.$
→Read the bar graph given in figure and answer the following questions:
$i.$ What is the information given by the bar graph?
$ii.$ What is the number of families having $6$ members?
$iii.$ How many members per family are there in the maximum number of families? Also tell the number of such families.
$iv.$ What are the number of members per family for which the number of families are equal? Also, tell the number of such families?
→$i.$ What is the information given by the bar graph?
$ii.$ What is the number of families having $6$ members?
$iii.$ How many members per family are there in the maximum number of families? Also tell the number of such families.
$iv.$ What are the number of members per family for which the number of families are equal? Also, tell the number of such families?
Find the mean of the following data:
→|
x:
|
$19$
|
$21$
|
$23$
|
$25$
|
$27$
|
$29$
|
$31$
|
|
f:
|
$13$
|
$15$
|
$16$
|
$18$
|
$16$
|
$15$
|
$13$
|
Draw the graph of the equations given below. Also, find the coordinates of the points where the graph cuts the coordinate axes: $6x - 3y = 12$
→In the given figure, $ABCD$ and $AEFG$ are two parallelograms. If $\angle\text{C}= 58^\circ,$ find $\angle\text{F}.$ 
→Find the value of $p$ for the following distribution whose mean is $16.6$.
→|
x:
|
$8$
|
$12$
|
$15$
|
$p$
|
$20$
|
$25$
|
$30$
|
|
f:
|
$12$
|
$16$
|
$20$
|
$24$
|
$16$
|
$8$
|
$4$
|