To do:We have to find the given products.Solution:Here, to find the given products we can use distributive property twice.Distributive Property:The distributive property of multiplication states that when a factor is multiplied by the sum or difference of two terms, it is essential to multiply each of the two numbers by the factor, and finally perform the addition or subtraction operation.$(a+b)(c+d)=a(c+d)+b(c+d)$..............(I)Therefore, (i) The given expression is $(x + 4) (x + 7)$.$(x + 4) (x + 7)=x(x+7)+4(x+7)$                      [Using (I)]$(x + 4) (x + 7)=x(x)+x(7)+4(x)+4(7)$$(x + 4) (x + 7)=x^2+7x+4x+28$$(x + 4) (x ... Read More

To do:We have to show that:(i) $(3x + 7)^2 - 84x = (3x - 7)^2$(ii) $(9a - 5b)^2 + 180ab = (9a + 5b)^2$(iii) $(\frac{4m}{3} - \frac{3n}{4})^2 + 2mn = \frac{16m^2}{9} + \frac{9n^2}{16}$(iv) $(4pq + 3q)^2 - (4pq - 3q)^2 = 48pq^2$(v) $(a - b) (a + b) + (b - c) (b + c) + (c - a) (c + a) = 0$Solution:To show that LHS $=$ RHS in each case, we can use the following algebraic identities:$(a+b)^2=a^2+2ab+b^2$.............(I)$(a-b)^2=a^2-2ab+b^2$.............(II)$(a+b)(a-b)=a^2-b^2$................(III)(i)  The given equation is $(3x + 7)^2 - 84x = (3x - 7)^2$Let us consider LHS, $(3x + 7)^2 - 84x ... Read More

Given:(i) $(x - y) (x + y) (x^2 + y^2) (x^4 + y^4)$(ii) $(2x - 1) (2x + 1) (4x^2 + 1) (16x^4 + 1)$(iii) $(7m - 8n)^2 + (7m + 8n)^2$(iv) $(2.5p - 1.5q)^2 - (1.5p - 2.5q)^2$(v) $(m^2 - n^2m)^2 + 2m^3n^2$To do:We have to simplify the given expressions.Solution:Here, we have to simplify the given expressions. By using the algebraic identities $(a+b)^2=a^2+2ab+b^2$,  $(a-b)^2=a^2-2ab+b^2$ and  $(a+b)(a-b)=a^2-b^2$, we can reduce the given expressions and simplify them. $(a+b)^2=a^2+2ab+b^2$.............(I)$(a-b)^2=a^2-2ab+b^2$.............(II)$(a+b)(a-b)=a^2-b^2$................(III)(i) The given expression is $(x - y) (x + y) (x^2 + y^2) (x^4 + y^4)$.$(x - y) (x + y) (x^2 + y^2) (x^4 + y^4)=(x^2-y^2)(x^2+y^2)(x^4+y^4)$  ... Read More

Given:The given expressions are(i) $4x^2 - 12x + 7$(ii) $4x^2 - 20x + 20$To do:We have to find the term that must be added to each of the given expression to make it a whole square.Solution:The given expressions are (i) $4x^2 - 12x + 7$ (ii) $4x^2 - 20x + 20$. Here, we have to find the term that must be added to each of the given expression to make it a whole square. So, to find the term that must be added, we have to make the given expressions as the sum of a whole square and some other term and using the ... Read More

Given:$x^2 + y^2 = 29$ and $xy = 2$To do:We have to find the value of(i) $x + y$(ii) $x - y$(iii) $x^4 + y^4$Solution:The given expressions are $x^2 + y^2 = 29$ and $xy = 2$. Here, we have to find the value of (i) $x + y$ (ii) $x - y$ (iii) $x^4 + y^4$. So, by using the identities $(a+b)^2=a^2+2ab+b^2$ and $(a-b)^2=a^2-2ab+b^2$, we can find the required values.$xy = 2$.........(I)$(a+b)^2=a^2+2ab+b^2$.............(II)$(a-b)^2=a^2-2ab+b^2$.............(III)(i) Let us consider, $x^2 + y^2 = 29$Adding $2xy$ on both sides, we get, $x^2+2xy+y^2=29+2xy$$(x+y)^2=29+2(2)$                    [Using (II) and (I)]$(x+y)^2=29+4$$(x+y)^2=33$Taking square root on both sides, ... Read More

Given:$2x + 3y = 14$ and $2x - 3y = 2$To do:We have to find the value of $xy$.Solution:The given expressions are $2x + 3y = 14$ and $2x - 3y = 2$. Here, we have to find the value of $xy$. So, by squaring and subtracting using the identities $(a+b)^2=a^2+2ab+b^2$ and $(a-b)^2=a^2-2ab+b^2$, we can find the required value.$(a+b)^2=a^2+2ab+b^2$.............(I)$(a-b)^2=a^2-2ab+b^2$.............(II)Let us consider, $2x + 3y = 14$Squaring on both sides, we get, $(2x + 3y)^2 = (14)^2$$(2x)^2+2(2x)(3y)+(3y)^2=196$            [Using (I)]$4x^2+12xy+9y^2=196$..........(III)Now, $2x - 3y = 2$Squaring on both sides, we get, $(2x - 3y)^2 = (2)^2$$(2x)^2-2(2x)(3y)+(3y)^2=4$            [Using (II)]$4x^2-12xy+9y^2=4$..........(IV)Subtracting ... Read More

Given:$x + \frac{1}{x} = 12$To do:We have to find the value of $x - \frac{1}{x}$.Solution:The given expression is $x + \frac{1}{x} = $. Here, we have to find the value of $x^4 + \frac{1}{x^4}$. So, by squaring the given expression and using the identities $(a+b)^2=a^2+2ab+b^2$ and $(a-b)^2=a^2-2ab+b^2$, we can find the required value.$(a+b)^2=a^2+2ab+b^2$.............(I)$(a-b)^2=a^2-2ab+b^2$.............(II)Let us consider, $x + \frac{1}{x} = 12$Squaring on both sides, we get, $(x + \frac{1}{x})^2 = (12)^2$$x^2+2\times x \times \frac{1}{x}+\frac{1}{x^2}=144$            [Using (I)]$x^2+2+\frac{1}{x^2}=144$$x^2+\frac{1}{x^2}=144-2$                       (Transposing $2$ to RHS)$x^2+\frac{1}{x^2}=142$Now, $x^2+\frac{1}{x^2}=142$Subtracting 2 from both sides, we get, $x^2+\frac{1}{x^2}-2=142-2$$x^2-2\times ... Read More

Given:$x + \frac{1}{x} = 9$To do:We have to find the value of $x^4 + \frac{1}{x^4}$.Solution:The given expression is $x + \frac{1}{x} = 9$. Here, we have to find the value of $x^4 + \frac{1}{x^4}$. So, by squaring the given expression and using the identity $(a+b)^2=a^2+2ab+b^2$, we can find the required value.$(a+b)^2=a^2+2ab+b^2$...................(i)Let us consider, $x + \frac{1}{x} = 9$Squaring on both sides, we get, $(x + \frac{1}{x})^2 = 9^2$$x^2+2\times x \times \frac{1}{x}+\frac{1}{x^2}=81$            [Using (I)]$x^2+2+\frac{1}{x^2}=81$$x^2+\frac{1}{x^2}=81-2$                       (Transposing $2$ to RHS)$x^2+\frac{1}{x^2}=79$Now, $x^2+\frac{1}{x^2}=79$Squaring on both sides, we get, $(x^2+\frac{1}{x^2})^2 = ... Read More

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

Given:$3x + 5y = 11$ and $xy = 2$To do:We have to find the value of $9x^2+25y^2$.Solution:The given expressions are $3x + 5y = 11$ and $xy = 2$. Here, we have to find the value of $9x^2+25y^2$. 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, $3x + 5y = 11$Squaring on both sides, we get, $(3x + 5y)^2 = (11)^2$                 [Using (ii)]$(3x)^2+2(3x)(5y)+(5y)^2=121$$9x^2+30xy+25y^2=121$$9x^2+30(2)+25y^2=121$                     [Using (i)]$9x^2+60+25y^2=121$$9x^2+25y^2=121-60$              (Transposing $60$ ... Read More