To make the given expression [tex]\( x^2 + 12x \)[/tex] a perfect square, follow these steps:
1. Identify the linear coefficient of [tex]\(x\)[/tex] in the expression. Here, the coefficient of [tex]\(x\)[/tex] is [tex]\(12\)[/tex].
2. To form a perfect square trinomial, take half of the linear coefficient and square it. In mathematical terms:
[tex]\[ \left( \frac{12}{2} \right)^2 \][/tex]
3. Calculate half of the linear coefficient:
[tex]\[ \frac{12}{2} = 6 \][/tex]
4. Now, square this result:
[tex]\[ 6^2 = 36 \][/tex]
5. Add this value to the original expression to form a perfect square trinomial:
[tex]\[ x^2 + 12x + 36 \][/tex]
Thus, the number you need to add to the expression [tex]\( x^2 + 12x \)[/tex] to make it a perfect square is [tex]\( 36 \)[/tex].
The perfect square trinomial obtained is:
[tex]\[ x^2 + 12x + 36 \][/tex]
Notice that this trinomial can be factored into the square of a binomial:
[tex]\[ (x + 6)^2 \][/tex]
So, the detailed solution is:
To the given expression [tex]\( x^2 + 12x \)[/tex], add [tex]\(36\)[/tex] to create the perfect square trinomial:
[tex]\[ x^2 + 12x + 36 \][/tex]
This trinomial can be written as:
[tex]\[ (x + 6)^2 \][/tex]
Hence, the number you should add is [tex]\( 36 \)[/tex], and the perfect square trinomial is [tex]\( x^2 + 12x + 36 \)[/tex].
