Sure, let's go through the process of completing the square for the equation [tex]\(x^2 + 8x - 3 = 0\)[/tex] step-by-step.
1. Start with the quadratic expression: [tex]\(x^2 + 8x - 3 = 0\)[/tex].
2. Focus on the quadratic and linear terms: [tex]\(x^2 + 8x\)[/tex]. We need to complete the square for this part.
3. Take the coefficient of the [tex]\(x\)[/tex] term (which is 8) and divide it by 2:
[tex]\[
\frac{8}{2} = 4
\][/tex]
4. Square this result:
[tex]\[
4^2 = 16
\][/tex]
5. Now, add and subtract this square (16) inside the equation to complete the square:
[tex]\[
x^2 + 8x = (x^2 + 8x + 16) - 16
\][/tex]
6. Rewrite the quadratic part as a perfect square and incorporate the constants:
[tex]\[
(x^2 + 8x + 16) - 16 - 3 = 0
\][/tex]
Simplify the constants:
[tex]\[
(x + 4)^2 - 19 = 0
\][/tex]
Therefore, when completing the square, [tex]\(x^2 + 8x - 3 = 0\)[/tex] can be transformed into:
[tex]\[
(x + 4)^2 - 19 = 0
\][/tex]
So, the left side of the equation becomes the binomial squared:
[tex]\[
(x + 4)^2
\][/tex]
And, to fill in the blanks in the steps:
[tex]\(
(x + 4)^2 = \ldots
\)[/tex]
The result from completing the square is:
[tex]\[
(x + 4)^2 - 19 = 0
\][/tex]
