Certainly! Let's work through the steps to rewrite the given equation [tex]\((5 + x)(5 - x) = 7\)[/tex] in its standard form [tex]\(Ax^2 + Bx + C = 0\)[/tex], and then identify the coefficients [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex].
1. First, expand the left-hand side of the equation:
[tex]\[
(5 + x)(5 - x)
\][/tex]
Recall the difference of squares formula [tex]\( (a + b)(a - b) = a^2 - b^2 \)[/tex]:
[tex]\[
(5 + x)(5 - x) = 5^2 - x^2 = 25 - x^2
\][/tex]
2. Replace the expanded form back into the equation:
[tex]\[
25 - x^2 = 7
\][/tex]
3. Move all terms to one side to obtain [tex]\(0\)[/tex] on the other side:
[tex]\[
25 - x^2 - 7 = 0
\][/tex]
Simplify the expression:
[tex]\[
18 - x^2 = 0
\][/tex]
4. Rearrange it to match the standard quadratic form [tex]\(Ax^2 + Bx + C = 0\)[/tex]:
[tex]\[
-x^2 + 18 = 0
\][/tex]
This can be written as:
[tex]\[
-x^2 + 0x + 18 = 0
\][/tex]
5. Identify the coefficients [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex]:
Comparing this with the standard form [tex]\(Ax^2 + Bx + C = 0\)[/tex]:
[tex]\[
A = -1
\][/tex]
[tex]\[
B = 0
\][/tex]
[tex]\[
C = 18
\][/tex]
Therefore, the correct values required to write the equation in standard form are [tex]\(A = -1\)[/tex], [tex]\(B = 0\)[/tex], and [tex]\(C = 18\)[/tex].
