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Sagot :
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].
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].
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