## If $x + \frac{1}{x} =20$, find the value of $x^2 + \frac{1}{x^2}$.

Updated on 01-Apr-2023 12:36:20
Given:$x + \frac{1}{x} =20$To do:We have to find the value of $x^2 + \frac{1}{x^2}$.Solution:The given expression is $x + \frac{1}{x} =20$. Here, we have to find the value of $x^2 + \frac{1}{x^2}$. So, by squaring the given expression and using the identity $(a+b)^2=a^2+2ab+b^2$, we can find the value of $x^2 + \frac{1}{x^2}$.$(a+b)^2=a^2+2ab+b^2$...................(i)Now, $x + \frac{1}{x} =20$Squaring on both sides, we get, $(x+\frac{1}{x})^2=(20)^2$$x^2+2\times x \times \frac{1}{x}+(\frac{1}{x})^2=400 [Using (i)]x^2+2+\frac{1}{x^2}=400$$x^2+\frac{1}{x^2}=400-2$                 (Transposing $2$ to RHS)$x^2+\frac{1}{x^2}=398$Hence, the value of $x^2+\frac{1}{x^2}$ is $398$.Read More

## Using the formula for squaring a binomial, evaluate the following:(i) $(102)^2$(ii) $(99)^2$(iii) $(1001)^2$(iv) $(999)^2$(v) $(703)^2$

Updated on 01-Apr-2023 12:32:28

## Find the values of the following expressions:(i) $16x^2 + 24x + 9$ when $x = \frac{7}{4}$(ii) $64x^2 + 81y^2 + 144xy$ when $x = 11$ and $y = \frac{4}{3}$(iii) $81x^2 + 16y^2 - 72xy$ when $x = \frac{2}{3}$ and $y = \frac{3}{4}$

Updated on 01-Apr-2023 12:24:10
To do:We have to find the values of the given expressions.Solution:Here, we have to find the values of the given expressions. So, simplifying the given expressions using the identities $(a+b)^2=a^2+2ab+b^2$.............(I) and $(a-b)^2=a^2-2ab+b^2$.............(II) and substituting the values of $x$ and $y$, we can find the required values.(i) The given expression is $16x^2 + 24x + 9$.$16x^2 + 24x + 9=(4x)^2+2\times 4x \times3+(3)^2$               [$24x=2\times 4x \times3$]$16x^2 + 24x + 9=(4x+3)^2$                (Using (I), $a=4x$ and $b=3$)Substituting $x = \frac{7}{4}$ in $(4x+3)^2$, we get, $(4x+3)^2=[4\times\frac{7}{4}+3]^2$ $(4x+3)^2=(7+3)^2$ $(4x+3)^2=(10)^2$ $(4x+3)^2=100$The value ... Read More

## If $3x + 5y = 11$ and $xy = 2$, find the value of $9x^2 + 25y^2$.

Updated on 01-Apr-2023 12:18:55