The formula [tex]\( c = \left(\frac{b}{2}\right)^2 \)[/tex] is used to complete the square in algebra.
Here is a detailed, step-by-step explanation:
1. Understanding "Completing the Square":
"Completing the square" is a method used in algebra to transform a quadratic expression of the form [tex]\( ax^2 + bx + c \)[/tex] into a perfect square trinomial. A perfect square trinomial is an expression that can be written as the square of a binomial.
2. Form of the Quadratic Expression:
Consider the quadratic expression [tex]\( x^2 + bx \)[/tex]. To complete the square, we need to transform it into the form [tex]\( (x + d)^2 \)[/tex], where [tex]\( d \)[/tex] is some number related to [tex]\( b \)[/tex].
3. Isolate the Terms:
Start with the expression [tex]\( x^2 + bx \)[/tex].
4. Determine the Value to Complete the Square:
To find the appropriate number to add and subtract (i.e., the value of [tex]\( c \)[/tex]), we use the formula [tex]\( c = \left(\frac{b}{2}\right)^2 \)[/tex]. This value of [tex]\( c \)[/tex] is what makes the expression a perfect square trinomial.
5. Applying the Formula:
Using the value obtained from [tex]\( c = \left(\frac{b}{2}\right)^2 \)[/tex]:
[tex]\[
x^2 + bx + \left(\frac{b}{2}\right)^2
\][/tex]
This trinomial can now be written as a perfect square:
[tex]\[
\left(x + \frac{b}{2}\right)^2
\][/tex]
6. Example:
For instance, if [tex]\( b = 6 \)[/tex]:
[tex]\[
c = \left(\frac{6}{2}\right)^2 = 3^2 = 9
\][/tex]
The original expression [tex]\( x^2 + 6x \)[/tex] can be transformed into:
[tex]\[
x^2 + 6x + 9
\][/tex]
Which can be written as:
[tex]\[
(x + 3)^2
\][/tex]
Thus, the formula [tex]\( c = \left(\frac{b}{2}\right)^2 \)[/tex] is used to complete the square in algebra by transforming a quadratic expression into a perfect square trinomial.
