Given:$x – y = 7$ and $xy = 9$To do:We have to find the value of $x^2+y^2$.Solution:The given expressions are $x – y = 7$ and $xy = 9$. Here, we have to find the value of $x^2 + y^2$. So, by squaring the given expression and using the identity $(a-b)^2=a^2-2ab+b^2$, we can find the required value.$xy = 9$............(i)$(a-b)^2=a^2-2ab+b^2$.............(ii)Now, $x – y = 7$Squaring on both sides, we get, $(x – y)^2 = 7^2$                 [Using (ii)]$x^2-2xy+y^2=49$$x^2-2(9)+y^2=49$                     [Using (i)]$x^2-18+y^2=49$$x^2+y^2=49+18$              ... Read More

Given:$x + y = 4$ and $xy = 2$To do:We have to find the value of $x^2 + y^2$.Solution:The given expressions are $x + y = 4$ and $xy = 2$. Here, we have to find the value of $x^2 + y^2$. So, by squaring the given expression and using the identity $(a+b)^2=a^2+2ab+b^2$, we can find the required value.$xy = 2$...........(i)$(a+b)^2=a^2+2ab+b^2$...........(ii)Now, $x + y = 4$Squaring on both sides, we get, $(x + y)^2 = 4^2$$x^2+2 \times x \times y+y^2=16$               [Using (ii)]$x^2+2xy+y^2=16$$x^2+2(2)+y^2=16$                          [Using ... Read More

Given:$x^2 + \frac{1}{x^2} = 18$To do:We have to find the values of $x + \frac{1}{x}$ and $x - \frac{1}{x}$.Solution:The given expression is $x^2 + \frac{1}{x^2} = 18$. Here, we have to find the values of $x + \frac{1}{x}$ and $x - \frac{1}{x}$. So, by using the identities $(a+b)^2=a^2+2ab+b^2$...................(i) and $(a-b)^2=a^2-2ab+b^2$.............(ii), we can find the required values.Now, $x^2 + \frac{1}{x^2} = 18$Adding $2$ on both sides, we get, $x^2 + \frac{1}{x^2} + 2 = 18+2$$x^2 + \frac{1}{x^2} + 2 \times x \times \frac{1}{x} = 20$               (Since $2\times x \times \frac{1}{x}=2$)$(x+\frac{1}{x})^2=20$            ... Read More

Given:$x - \frac{1}{x} = 3$To do:We have to find the values of $x^2 + \frac{1}{x^2}$ and $x^4 + \frac{1}{x^4}$.Solution:The given expression is $x - \frac{1}{x} = 3$. Here, we have to find the values of $x^2 + \frac{1}{x^2}$ and $x^4 + \frac{1}{x^4}$. So, by squaring the given expression and using the identities $(a+b)^2=a^2+2ab+b^2$...................(i) and $(a-b)^2=a^2-2ab+b^2$.............(ii), we can find the required values. Let us consider, $x - \frac{1}{x} = 3$Squaring on both sides, we get, $(x - \frac{1}{x})^2 = 3^2$                 [Using (ii)]$x^2-2\times x \times \frac{1}{x}+\frac{1}{x^2}=9$$x^2-2+\frac{1}{x^2}=9$$x^2+\frac{1}{x^2}=9+2$                     (Transposing $-2$ ... Read More

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

Given:(i) $4x = (52)^2 – (48)^2$  (ii) $14x = (47)^2 – (33)^2$  (iii) $5x = (50)^2 – (40)^2$To do:We have to find the value of $x$ in each case.Solution:Here, we have to find the value of $x$ in each expression. The given expressions are the difference of two squares. So, to find the value of $x$ we can simplify the RHS in each case using the identity:$(a – b) (a + b) = a^2 – b^2$.Therefore, (i) $4x = (52)^2 – (48)^2$ This implies, $4x=(52+48)\times(52-48)$$4x=100\times4$$4x=400$$x=\frac{400}{4}$$x=100$Hence, the value of $x$ is $100$.(ii) $14x = (47)^2 – (33)^2$ This implies, $14x=(47+33)\times(47-33)$$14x=80\times14$$x=\frac{80\times14}{14}$$x=80$Hence, the value of $x$ ... Read More

Given:(i) $(82)^2 – (18)^2$  (ii) $(467)^2 – (33)^2$  (iii) $(79)^2 – (69)^2$  (iv) $197 \times 203$  (v) $113 \times 87$  (vi) $95 \times 105$  (vii) $1.8 \times 2.2$  (viii) $9.8 \times 10.2$To do:We have to simplify the given expressions using the formula: $(a – b) (a + b) = a^2 – b^2$Solution:Here, we have to simplify the given expressions using the formula $(a – b) (a + b) = a^2 – b^2$. The given expressions can be written as the difference of two squares by writing the terms as the sum or difference of two suitable numbers.  (i) The given expression is $(82)^2 – ... Read More

Given:(i) $\frac{((58)^2 – (42)^2)}{16}$  (ii) $178 \times 178 – 22 \times 22$  (iii) $\frac{(198 \times 198 – 102 \times 102)}{96}$  (iv) $1.73 \times 1.73 – 0.27 \times 0.27$  (v) $\frac{(8.63 \times 8.63 – 1.37 \times 1.37)}{0.726}$To do:We have to simplify the given expressions using suitable identities.Solution:Here, we have to simplify the given expressions. The given expressions(numerators in the expressions) are in the form of difference of two square numbers. We can simplify the given expressions by using the identity $a^2-b^2=(a+b) \times (a-b)$.(i) The given expression is $\frac{((58)^2 – (42)^2)}{16}$ Here, $a=58$ and $b=42$Therefore, $\frac{((58)^2 – (42)^2)}{16}=\frac{(58+42) \times (58-42)}{16}$$\frac{((58)^2 – (42)^2)}{16}=\frac{100\times16}{16}$$\frac{((58)^2 – (42)^2)}{16}=100$Hence, ... Read More

To do:We have to evaluate the given expressions using the formula for squaring a binomial.Solution:Here, we have to find the squares of some large numbers. We can find the squares of multiples of $10^n$ easily. So, express the given numbers as the sum of multiples of $10^n$ and other numbers. We can then find the squares of the given numbers by expanding the squares using the algebraic expressions:$(a+b)^2 = a^2+2ab+b^2$$(a-b)^2 = a^2-2ab+b^2$(i) The given expression is $(102)^2$.$102$ can be written as $100+2$We know that, $(a+b)^2 = a^2+2ab+b^2$Here, $a=100$ and $b=2$Therefore, $(100+2)^2=(100)^2+2\times100\times2+2^2$$(100+2)^2=10000+400+4$$(100+2)^2=10404$Hence, $(102)^2=10404$.(ii) The given expression is $(99)^2$.$99$ can be written as $100-1$We know ... Read More

2D shapes are flat with only length and breadth, while 3D shapes are solid objects with length, breadth, and height. In this brief article, we will take a look at the features of 2D and 3D shapes and identify how they differ from each other.2D ShapesA 2D shape has two dimensions, that is, Length and Breadth. 2D shapes are flat because they don't have any height or depth. Examples of 2D shapes include circle, rectangle, square, polygons, etc.Since 2D shapes don't have any height, they don't have any volume either. 2D shapes have only areas. 2D shapes are drawn using ... Read More