# Algebraic identity problems – Grade 09

Given below is the collection of questions related to algebraic identities with detailed solution.

All the questions are to the standard of grade 9.

Question 01
Solve the below expression;
\mathtt{\left( 2x-\frac{1}{x}\right)^{2}}

Solution
Referring the formula;
\mathtt{( a-b)^{2} =a^{2} +b^{2} -2ab}

Using the formula;

\mathtt{\Longrightarrow \ ( 2x)^{2} +\left(\frac{1}{x}\right)^{2} -2.2x.\frac{1}{x}}\\\ \\ \mathtt{\Longrightarrow \ 4x^{2} +\frac{1}{x^{2}} -4}

Hence, the above expression is the solution.

Question 02
Solve the below expression;
\mathtt{( 2x+y)( 2x-y)}\

Solution
Referring the formula;
\mathtt{( a+b)( a-b) \ =\ a^{2} -b^{2}}

Using the formula we get;

\mathtt{\Longrightarrow \ ( 2x)^{2} -y^{2}}\\\ \\ \mathtt{\Longrightarrow \ 4x^{2} -y^{2}}

Hence, the above expression is the solution.

Question 03
Solve the below expression;
\mathtt{\left( x^{2} y+y^{2} x\right)^{2}}

Solution
Referring the formula;
\mathtt{( a+b)^{2} =a^{2} +b^{2} +2ab}

Using the formula, we get;

\mathtt{\Longrightarrow \ \left( x^{2} y\right)^{2} +\left( y^{2} x\right)^{2} +2\left( x^{2} y\right)\left( y^{2} x\right)}\\\ \\ \mathtt{\Longrightarrow \ x^{4} y^{2} +x^{2} y^{4} +2x^{3} y^{3}}

Hence, the above expression is the solution.

Question 04
Find the value of number \mathtt{( 399)^{2}} using algebraic entities.

Solution
The above number can be written as;

\mathtt{\Longrightarrow ( 399)^{2}}\\\ \\ \mathtt{\Longrightarrow ( 400-1)^{2}}

Now referring the formula;
\mathtt{( a-b)^{2} =a^{2} +b^{2} -2ab}

Using the above formula, we get;

\mathtt{\Longrightarrow ( 400-1)^{2} \ }\\\ \\ \mathtt{\Longrightarrow \ 400^{2} +1^{2} -2( 400)( 1)}\\\ \\ \mathtt{\Longrightarrow \ 160000+1-800}\\\ \\ \mathtt{\Longrightarrow \ 159201}

Hence, 159201 is the solution.

Question 05
Find the value of below expression using algebraic entity
117 x 83

Solution
The above number can be written as;

\mathtt{\Longrightarrow 117\times 83}\\\ \\ \mathtt{\Longrightarrow \ ( 100+17) \times ( 100-17)}

Referring the formula;
\mathtt{( a+b)( a-b) \ =\ a^{2} -b^{2}}

Using the formula, we get;

\mathtt{\Longrightarrow \ ( 100+17) \times ( 10-17)}\\\ \\ \mathtt{\Longrightarrow \ 100^{2} -17^{2}}\\\ \\ \mathtt{\Longrightarrow \ 10000-289}\\\ \\ \mathtt{\Longrightarrow \ 9711}

Hence, 9711 is the solution.

Question 06
If a + b = 10 and ab = 21;
then find the value of \mathtt{a^{3} +b^{3}}

Solution
Referring to cube of sum formula;
\mathtt{( a+b)^{3} =a^{3} +b^{3} +3ab( a+b)}

Using the formula, we get;

\mathtt{( a+b)^{3} =a^{3} +b^{3} +3ab( a+b)}\\\ \\ \mathtt{10^{3} \ =\ a^{3} +b^{3} +3\times 21( 10)}\\\ \\ \mathtt{10^{3} =a^{3} +b^{3} +630}\\\ \\ \mathtt{a^{3} +b^{3} \ =\ 1000-630}\\\ \\ \mathtt{a^{3} +b^{3} =\ 370}

Hence, 370 is the right answer.

Next chapter : Problems on algebraic identities – se

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