To convert the given quadratic equation [tex]\(x^2 - 14x + 18 = 0\)[/tex] into the form [tex]\((x - p)^2 = q\)[/tex], we need to complete the square. Here are the detailed steps:
1. Start with the original equation:
[tex]\[
x^2 - 14x + 18 = 0
\][/tex]
2. Move the constant term to the other side of the equation:
[tex]\[
x^2 - 14x = -18
\][/tex]
3. Complete the square on the left-hand side. To do this, we need to add and subtract a specific value to convert the left-hand side into a perfect square trinomial. That value is determined by taking half of the coefficient of [tex]\(x\)[/tex] (which is [tex]\(-14\)[/tex]), squaring it, and then adding and subtracting this square within the equation.
The coefficient of [tex]\(x\)[/tex] is [tex]\(-14\)[/tex]. Half of [tex]\(-14\)[/tex] is [tex]\(-7\)[/tex], and squaring [tex]\(-7\)[/tex] gives us [tex]\(49\)[/tex].
4. Add and subtract [tex]\(49\)[/tex] on the left-hand side:
[tex]\[
x^2 - 14x + 49 - 49 = -18
\][/tex]
5. Rewrite this equation to group the perfect square trinomial:
[tex]\[
(x^2 - 14x + 49) - 49 = -18
\][/tex]
6. Simplify by combining the constants [tex]\(-49\)[/tex] and [tex]\(-18\)[/tex] on the right-hand side:
[tex]\[
(x - 7)^2 - 49 = -18
\][/tex]
7. Move [tex]\(-49\)[/tex] to the right-hand side to isolate the perfect square:
[tex]\[
(x - 7)^2 = 31
\][/tex]
Hence, the quadratic equation [tex]\(x^2 - 14x + 18 = 0\)[/tex] is rewritten in the form [tex]\((x - p)^2 = q\)[/tex] with [tex]\(p = 7\)[/tex] and [tex]\(q = 31\)[/tex].
