Question
If $\text{x}=2+\sqrt3,$ find the value of $\text{x}+\frac{1}{\text{x}}.$
✓
Answer
Given that $\text{x}=2+\sqrt3,$ Hence $\frac{1}{\text{x}}$ is given as$\frac{1}{\text{x}}=\frac{1}{2+\sqrt3}.$ we are asked to find $\text{x}+\frac{1}{\text{x}}$
We know that rationalization factor for $2+\sqrt3$ is $2-\sqrt3.$ We will multiply each side of the given expression $\frac{1}{2+\sqrt3}$ by $2-\sqrt3,$ to get$\frac{1}{\text{x}}=\frac{1}{2+\sqrt3}\times\frac{2-\sqrt3}{2-\sqrt3}$
$=\frac{2-\sqrt3}{(2)^2-\big(\sqrt3\big)^2}$
$=\frac{2-\sqrt3}{4-3}$
$=2-\sqrt3$
Therefore,$\text{x}+\frac{1}{\text{x}}=2+\sqrt3+2-\sqrt3$
$=4$
Hence the value of the given expresion is 4.
We know that rationalization factor for $2+\sqrt3$ is $2-\sqrt3.$ We will multiply each side of the given expression $\frac{1}{2+\sqrt3}$ by $2-\sqrt3,$ to get$\frac{1}{\text{x}}=\frac{1}{2+\sqrt3}\times\frac{2-\sqrt3}{2-\sqrt3}$
$=\frac{2-\sqrt3}{(2)^2-\big(\sqrt3\big)^2}$
$=\frac{2-\sqrt3}{4-3}$
$=2-\sqrt3$
Therefore,$\text{x}+\frac{1}{\text{x}}=2+\sqrt3+2-\sqrt3$
$=4$
Hence the value of the given expresion is 4.
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
Ratio of two adjacent sides of a parallelogram is 3 : 4, and its perimeter is 112 cm. Find the length of its each side.
In the given figure, O and O' are centers of two circles intersecting at B and C. ACD is a straight line, find x.
A child playing with building blocks, which are of the shape of the cubes, has built a structure as shown in Figure.
If the edge of each cube is $3\ cm$, find the volume of the structure built by the child.
→If the edge of each cube is $3\ cm$, find the volume of the structure built by the child.
An equilateral triangle of side 9cm is inscribed in a circle. Find the radius of the circle.
In the given figure, ABCD is a trapezium. Find the values of x and y.
In the figure given below, AB || CD. Find the value of x in each case.
Two cubes, each of volume $512 cm^3$ are joined end to end. Find the surface area of the resulting cuboid.
→If the mean of the following data is 20.2, then find the value of p.
P is a point on the bisector of $\angle\text{ABC}.$ If the line through P, parallel to BA meets BC at Q, prove that $\triangle\text{BPQ}$ is an isosceles triangle.
→The Blood group table of 30 students of class IX is recorded as follows:
A, B, O, O, AB, O, A, O, B, A, O, B, A, O, O
A, AB, O, A, A, O, O, AB, B, A, O, B, A, B, O
A student is selected at random from the class from blood donation. Find the probability that the blood group of the student chosen is:
A, B, O, O, AB, O, A, O, B, A, O, B, A, O, O
A, AB, O, A, A, O, O, AB, B, A, O, B, A, B, O
A student is selected at random from the class from blood donation. Find the probability that the blood group of the student chosen is:
- A
- B
- AB
- O