Sure! Let's solve the equation [tex]\( x^2 - 16x + 60 = -12 \)[/tex] by completing the square step by step.
First, we need to isolate the quadratic and linear terms on one side of the equation.
Step 1: Subtract 60 from each side of the equation.
[tex]\[ x^2 - 16x + 60 - 60 = -12 - 60 \][/tex]
This simplifies to:
[tex]\[ x^2 - 16x = -72 \][/tex]
Next, we will complete the square on the left-hand side of the equation. To do this, we need to add a specific value to both sides of the equation that allows the left-hand side to be written as a perfect square trinomial.
Step 2: To complete the square, we add [tex]\(\left(\frac{b}{2}\right)^2\)[/tex] to each side of the equation, where [tex]\( b \)[/tex] is the coefficient of [tex]\( x \)[/tex].
In this case, [tex]\( b = -16 \)[/tex].
Calculate the term to add:
[tex]\[ \left(\frac{-16}{2}\right)^2 = (-8)^2 = 64 \][/tex]
So, we add 64 to both sides of the equation:
[tex]\[ x^2 - 16x + 64 = -72 + 64 \][/tex]
This simplifies to:
[tex]\[ x^2 - 16x + 64 = -8 \][/tex]
Now the equation is in the form of a completed square.
So, the values we found during the process are:
- The value we subtracted from both sides initially: [tex]\( -72 \)[/tex]
- The value we added to complete the square: [tex]\( 64 \)[/tex]
- The final value on the right-hand side of the equation: [tex]\( -8 \)[/tex]
