In this problem, we have to complete the square for a quadratic equation.
Recall that a perfect square trinomial comes in the form:
[tex]a^2+2ab+b^2[/tex]
Sometimes, we have to create that form by using the following method to complete the square:
1. Get the equation equal to
[tex]ax^2+bx=-c[/tex]
2. Divide the b-term and square it:
[tex](\frac{b}{2})^2[/tex]
3. Add that new value to both sides of the equation:
[tex]ax^2+bx+(\frac{b}{2})^2=-c+(\frac{b}{2})^2[/tex]
It looks really confusing in this format, so let's follow our equation to get a better idea.
We are given:
[tex]x^2-6x=7[/tex]
Luckily, we already have it in the format required for Step 1. So we can complete Step 2 by identifying the b-value.
[tex]\begin{gathered} x^2-6x=7 \\ \\ \text{ The b-value is:}b=-6 \end{gathered}[/tex]
So, we have:
[tex](\frac{-6}{2})^\rightarrow(-3)^{^2}\rightarrow9[/tex]
Finally for Step 3, we add that 9 to both sides to get:
[tex]\begin{gathered} x^2-6x+9=7+9 \\ \\ x^2-6x+9=16 \end{gathered}[/tex]
The term tha tmust be added to the equation to make it into a perfect square is 9.
