"Isolate the constant by adding 7 to both sides of the equation."
This step separates the non-squareable 7 and the squareable [tex] x^2 - 6x [/tex] .
"Add 9 to both sides of [tex] x^2 - 6x = 7 [/tex] to form a perfect square trinomial while keeping the equation balanced."
After separating the non-squareable, add the number which makes the first or left side a perfect square trinomial. The formula to find the number is: [tex] c = {\frac{b}{2} }^{2} [/tex] .
When we plug the values: [tex] c = { \frac{ - 6}{2} }^{2} [/tex]
Simplify: [tex] c = {( - 3)}^{2} = 9 [/tex]
"Write the trinomial [tex] x^2 - 6x + 9 [/tex] as [tex] (x - 3) [/tex] squared."
When you factor [tex] x^2 - 6x + 9 [/tex] , you will get [tex] (x - 3) \times (x - 3) [/tex] .
"Use the square root property of equality to get [tex] x - 3 = ± \sqrt{16} [/tex] ."
The 16 is coming from the part when we add 9. We needed 9 on the left side for a perfect square, but to protect the balance of the equality, we need to add 9 to the right side too. When we add 7 and 9, we got 16, and that is where it came from.
"Isolate the variable x to get solutions of -1 and 7."
To isolate x we branched the plus-minus sign:
[tex]x - 3 = 4 \: \: \: x = 4 + 3 = 7 \\ \\ x - 3 = - 4 \: \: \: x = - 4 + 3 = - 1[/tex]
