Question

How do you write \[{x^2} + 3x\] in factored form?

Answer 1

Hint:Here, we are required to factorize the given polynomial. We will use the method of ‘Completing the square’ to expand the given polynomial. By using the algebraic formulas of the square of the sum of two numbers and the difference of the square of two numbers we will get the required factors.

Formula Used:
\[{a^2} + 2ab + {b^2} = {\left( {a + b} \right)^2}\]
\[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\]

Complete step-by-step answer:
In order to write the given quadratic equation \[{x^2} + 3x\] in factored form, we will use the method of ‘Completing the square’ to expand this polynomial.
This means that we have to change this polynomial in the form of \[{a^2} + 2ab + {b^2}\] by adding and subtracting terms such that the given polynomial remains the same.
Hence, by completing the square, we can write the given polynomial as:
\[{x^2} + 3x = {\left( x \right)^2} + 2\left( x \right)\left( {\dfrac{3}{2}} \right) + {\left( {\dfrac{3}{2}} \right)^2} - {\left( {\dfrac{3}{2}} \right)^2}\]
If we observe this carefully, then LHS is equal to RHS just the way of writing the polynomial has changed.
Now, we will use the formula \[{a^2} + 2ab + {b^2} = {\left( {a + b} \right)^2}\] in the above equation. Therefore, we get
\[ \Rightarrow {x^2} + 3x = {\left( {x + \dfrac{3}{2}} \right)^2} - {\left( {\dfrac{3}{2}} \right)^2}\]
Now, using the formula \[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\], we can write the above RHS as:
\[ \Rightarrow {x^2} + 3x = \left( {x + \dfrac{3}{2} + \dfrac{3}{2}} \right)\left( {x + \dfrac{3}{2} - \dfrac{3}{2}} \right)\]
Adding and subtracting the like terms, we get
\[ \Rightarrow {x^2} + 3x = \left( {x + 3} \right)x\]

Hence, clearly, the factors of the given polynomial \[{x^2} + 3x\] are: \[x\] and \[\left( {x + 3} \right)\].

Hence, this is the required answer.

Note: An alternate way to solve this question is the direct method of a greatest common factor.
In this method, we directly take out the greatest factor which is common from the polynomial and leave the rest of the terms inside the bracket. The terms in the bracket become one factor and the common terms were taken out to become the other factor of the given polynomial.
Hence, given polynomial is: \[{x^2} + 3x\]
The greatest common factor is \[x\].
Hence, taking it out from the bracket, we get,
\[{x^2} + 3x = x\left( {x + 3} \right)\]
Hence, the factors of the given polynomial are \[x,\left( {x + 3} \right)\].