What number should be added to both sides of the equation to complete the square?

[tex]\[ x^2 + 3x = 6 \][/tex]

A. \(\frac{3}{2}\)

B. \(\left(\frac{3}{2}\right)^2\)

C. 3

D. [tex]\(6^2\)[/tex]



Answer :

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]