To determine what number should be added to both sides of the equation \( x^2 + 3x = 6 \) to complete the square, we will follow these steps:
1. Identify the coefficient of \( x \):
The given equation is \( x^2 + 3x = 6 \). The coefficient of \( x \) is \( 3 \).
2. Divide the coefficient of \( x \) by 2:
We take the coefficient of \( x \), which is \( 3 \), and divide it by \( 2 \):
[tex]\[
\frac{3}{2}
\][/tex]
3. Square the result:
Take the result from the previous step and square it:
[tex]\[
\left( \frac{3}{2} \right)^2
\][/tex]
4. Simplify the square:
Simplify the expression \( \left( \frac{3}{2} \right)^2 \):
[tex]\[
\left( \frac{3}{2} \right)^2 = \frac{3 \times 3}{2 \times 2} = \frac{9}{4}
\][/tex]
Thus, the number that should be added to both sides of the equation \( x^2 + 3x = 6 \) to complete the square is \( \frac{9}{4} \) or \( 2.25 \).
So the correct answer is:
[tex]\[
\boxed{\left( \frac{3}{2} \right)^2}
\][/tex]
To verify, we can transform the equation with the added number to a perfect square trinomial:
[tex]\[
x^2 + 3x + \frac{9}{4} = 6 + \frac{9}{4}
\][/tex]
This can be written as:
[tex]\[
\left( x + \frac{3}{2} \right)^2 = \frac{33}{4}
\][/tex]
