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# If $x + y = \dfrac{7}{2}$ and $xy = \dfrac{5}{2}$, find the value of:(a) $x - y$(b) ${x^2} - {y^2}$

Last updated date: 18th Jun 2024
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Hint: Here, we have to find the value of $x - y$ and ${x^2} - {y^2}$. We will use the algebraic identity for the sum of two numbers to find the sum of squares of $x$ and $y$. Then, using the algebraic identity for the square of difference of two numbers, we will find the value of $x - y$. Finally, using the algebraic identity for the product of the sum and difference of two numbers, we will find the value of ${x^2} - {y^2}$.

Formula Used: The square of the sum of two numbers $a$ and $b$ is given by the algebraic identity ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$.
The square of the difference of two numbers $a$ and $b$ is given by the algebraic identity ${\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab$.
The product of the sum and difference of two numbers $a$ and $b$ is given by the algebraic identity ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$.

(a)
We know that the square of the sum of two numbers $a$ and $b$ is given by the algebraic identity ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$.
Using the algebraic identity, we get the square of the sum of the numbers $x$ and $y$ as
${\left( {x + y} \right)^2} = {x^2} + {y^2} + 2xy$
Substituting $x + y = \dfrac{7}{2}$ and $xy = \dfrac{5}{2}$ in the equation, we get
$\Rightarrow {\left( {\dfrac{7}{2}} \right)^2} = {x^2} + {y^2} + 2\left( {\dfrac{5}{2}} \right)$
Simplifying the expression, we get
$\Rightarrow \dfrac{{49}}{4} = {x^2} + {y^2} + 5$
Subtracting 5 from both sides, we get
$\begin{array}{l} \Rightarrow \dfrac{{49}}{4} - 5 = {x^2} + {y^2} + 5 - 5\\ \Rightarrow \dfrac{{49 - 20}}{4} = {x^2} + {y^2}\\ \Rightarrow \dfrac{{29}}{4} = {x^2} + {y^2}\end{array}$
Using the algebraic identity, we get the square of the difference of the numbers $x$ and $y$ as
${\left( {x - y} \right)^2} = {x^2} + {y^2} - 2xy$
Substituting ${x^2} + {y^2} = \dfrac{{29}}{4}$ and $xy = \dfrac{5}{2}$ in the equation, we get
$\Rightarrow {\left( {x - y} \right)^2} = \dfrac{{29}}{4} - 2\left( {\dfrac{5}{2}} \right)$
Simplifying the equation, we get
$\Rightarrow {\left( {x - y} \right)^2} = \dfrac{{29}}{4} - 5$
Subtracting the terms of the expression, we get
$\begin{array}{l} \Rightarrow {\left( {x - y} \right)^2} = \dfrac{{29 - 20}}{4}\\ \Rightarrow {\left( {x - y} \right)^2} = \dfrac{9}{4}\end{array}$
Applying the square root of both sides, we get
$\Rightarrow x - y = \sqrt {\dfrac{9}{4}} = \dfrac{3}{2}$
Therefore, the value of $x - y$ is $\dfrac{3}{2}$.
(b)
We will use the algebraic identity for the product of the sum and difference of two numbers to find the value of ${x^2} - {y^2}$.
We know that the product of the sum and difference of two numbers $a$ and $b$ is given by the algebraic identity ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$.
Using the algebraic identity, we get the product of the sum and difference of the numbers $x$ and $y$ as
${x^2} - {y^2} = \left( {x + y} \right)\left( {x - y} \right)$
Substituting $x + y = \dfrac{7}{2}$ and $x - y = \dfrac{3}{2}$ in the equation, we get
$\Rightarrow {x^2} - {y^2} = \left( {\dfrac{7}{2}} \right)\left( {\dfrac{3}{2}} \right)$
Multiplying the terms of the expression, we get
$\Rightarrow {x^2} - {y^2} = \dfrac{{21}}{4}$
Therefore, the value of ${x^2} - {y^2}$ is $\dfrac{{21}}{4}$.

Note: We have assumed that $x > y$. The answer will be different if $x < y$.
Taking the square root of both sides ${\left( {x - y} \right)^2} = \dfrac{9}{4}$, we get
$\begin{array}{l} \Rightarrow x - y = - \sqrt {\dfrac{9}{4}} \\ \Rightarrow x - y = - \dfrac{3}{2}\end{array}$
The negative sign is taken because it is assumed that $x < y$.
Substituting $x + y = \dfrac{7}{2}$ and $x - y = - \dfrac{3}{2}$ in the equation ${x^2} - {y^2} = \left( {x + y} \right)\left( {x - y} \right)$, we get
$\Rightarrow {x^2} - {y^2} = \left( {\dfrac{7}{2}} \right)\left( { - \dfrac{3}{2}} \right)$
Multiplying the terms of the expression, we get
$\Rightarrow {x^2} - {y^2} = - \dfrac{{21}}{4}$
Therefore, if $x < y$, then the values of $x - y$ and ${x^2} - {y^2}$ will be $- \dfrac{3}{2}$ and $- \dfrac{{21}}{4}$ respectively.