Factorization of algebraic expression – Problem set 03

Given below are question related to factorization of algebraic expression with detailed solution.

All the questions are to the standard of grade 09.

Question 01
Factorize the below expression;
\mathtt{2x^{2} +2\sqrt{6} xy+3y^{2}}

Solution
The above expression can be written as;

\mathtt{\Longrightarrow \ 2x^{2} +2\sqrt{6} xy+3y^{2}}\\\ \\ \mathtt{\Longrightarrow \ \left(\sqrt{2} x\right)^{2} +2.\sqrt{2} x.\sqrt{3} y+\left(\sqrt{3} y\right)^{2}}


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

Using the formula, we get;
\mathtt{\Longrightarrow \ \left(\sqrt{2} x+\sqrt{3} y\right)^{2}}

Hence, the above expression is the solution.

Question 02
Factorize the below expression;
\mathtt{x^{2} +6\sqrt{2} x+10}

Solution
The above expression can be written as;

\mathtt{\Longrightarrow \ x^{2} +6\sqrt{2} x+10}\\\ \\ \mathtt{\Longrightarrow \ x^{2} +5\sqrt{2} x+\sqrt{2} x+10}\\\ \\ \mathtt{\Longrightarrow \ x^{2} +5\sqrt{2} x+\sqrt{2} x+5\sqrt{2} .\sqrt{2}}

Separating the common terms.


\mathtt{\Longrightarrow \ x\left( x+5\sqrt{2}\right) +\sqrt{2}\left( x+5\sqrt{2}\right)}\\\ \\ \mathtt{\Longrightarrow \ \left( x+\sqrt{2}\right)\left( x+5\sqrt{2}\right)}

Hence, the above expression is the solution.

Question 03
Factorize the below expression;
\mathtt{x^{2} -\sqrt{3} x-6}

Solution
\mathtt{\Longrightarrow \ x^{2} -\sqrt{3} x-6}\\\ \\ \mathtt{\Longrightarrow \ x^{2} -2\sqrt{3} x+\sqrt{3} x-6}\\\ \\ \mathtt{\Longrightarrow \ x^{2} -2\sqrt{3} x+\sqrt{3} x-2\sqrt{3}\sqrt{3}}\\\ \\ \mathtt{\Longrightarrow \ x\left( x-2\sqrt{3}\right) +\sqrt{3}\left( x-2\sqrt{3}\right)}\\\ \\ \mathtt{\Longrightarrow \ \left( x+\sqrt{3}\right)\left( x-2\sqrt{3}\right)}

Hence, the above expression is the solution.

Question 04
Factorize the below expression;
\mathtt{x^{2} +5\sqrt{5} x+30}

Solution
\mathtt{\Longrightarrow \ x^{2} +5\sqrt{5} x+30}\\\ \\ \mathtt{\Longrightarrow \ x^{2} +3\sqrt{5} x+2\sqrt{5} x+30}\\\ \\ \mathtt{\Longrightarrow \ x\left( x+3\sqrt{5}\right) +2\sqrt{5}\left( x+3\sqrt{5}\right)}\\\ \\ \mathtt{\Longrightarrow \ \left( x+2\sqrt{5}\right)\left( x+3\sqrt{5}\right)}

Hence, the above expression is the solution.

Question 05
Factorize the below algebraic expression.
\mathtt{2x^{2} +3\sqrt{5} x+5}

Solution
\mathtt{\Longrightarrow \ 2x^{2} +3\sqrt{5} x+5}\\\ \\ \mathtt{\Longrightarrow 2x^{2} +2\sqrt{5} x+\sqrt{5} x+5}\\\ \\ \mathtt{\Longrightarrow \ 2x\left( x+\sqrt{5}\right) +\sqrt{5}\left( x+\sqrt{5}\right)}\\\ \\ \mathtt{\Longrightarrow \ \left( 2x+\sqrt{5}\right)\left( x+\sqrt{5}\right)}

Hence, the above expression is the solution.

Question 06
Factorize the below algebraic expression;
\mathtt{\ 5\sqrt{5} x^{2} +20x+3\sqrt{5}}

Solution
\mathtt{\Longrightarrow \ 5\sqrt{5} x^{2} +20x+3\sqrt{5}}\\\ \\ \mathtt{\Longrightarrow \ 5\sqrt{5} x^{2} +5x+15x+3\sqrt{5}}\\\ \\ \mathtt{\Longrightarrow \ \sqrt{5} x\ \left( 5x+\sqrt{5}\right) +3\left( 5x+\sqrt{5}\right)}\\\ \\ \mathtt{\Longrightarrow \ \left(\sqrt{5} x+3\right)\left( 5x+\sqrt{5}\right)}

The above expression is the solution.

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