To solve the quadratic equation [tex]\( x^2 + 6x - 7 = 0 \)[/tex] by completing the square, we need to follow these steps:
1. Start with the given equation:
[tex]\[
x^2 + 6x - 7 = 0
\][/tex]
2. Isolate the terms containing [tex]\( x \)[/tex] on one side of the equation:
Add 7 to both sides to balance the equation:
[tex]\[
x^2 + 6x = 7
\][/tex]
3. Find the value [tex]\( c \)[/tex] that completes the square:
To complete the square, we need to make the left side of the equation a perfect square trinomial. The perfect square trinomial takes the form [tex]\((x + b)^2\)[/tex], where [tex]\( b \)[/tex] is half the coefficient of [tex]\( x \)[/tex].
The coefficient of [tex]\( x \)[/tex] is 6. We take half of this coefficient:
[tex]\[
\frac{6}{2} = 3
\][/tex]
Next, square this result:
[tex]\[
3^2 = 9
\][/tex]
4. Add [tex]\( c \)[/tex] to both sides of the equation:
The value of [tex]\( c \)[/tex] is 9, so add it to both sides:
[tex]\[
x^2 + 6x + 9 = 7 + 9
\][/tex]
5. Verify the value of [tex]\( c \)[/tex]:
It is clear that adding 9 completes the square.
Therefore, the value of [tex]\( c \)[/tex] is:
[tex]\[
c = 9
\][/tex]
