To determine which equation has an [tex]\(a\)[/tex]-value of 1, a [tex]\(b\)[/tex]-value of -3, and a [tex]\(c\)[/tex]-value of -5, we need to analyze each given equation and rearrange it, if necessary, to the standard form of a quadratic equation: [tex]\(0 = ax^2 + bx + c\)[/tex].
Let's examine each equation one by one:
1. [tex]\(0 = -3x - 5 + x^2\)[/tex]
Rearrange the terms to match the standard quadratic form:
[tex]\[0 = x^2 - 3x - 5\][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = -3\)[/tex], and [tex]\(c = -5\)[/tex]. These coefficients match the required values. Hence, this equation is a candidate.
2. [tex]\(0 = x - 3 - 5x^2\)[/tex]
Rearrange the terms to match the standard quadratic form:
[tex]\[0 = -5x^2 + x - 3\][/tex]
Here, [tex]\(a = -5\)[/tex], [tex]\(b = 1\)[/tex], and [tex]\(c = -3\)[/tex]. These coefficients do not match the required values. Therefore, this equation does not meet the criteria.
3. [tex]\(0 = 3x - 5 - x^2\)[/tex]
Rearrange the terms to match the standard quadratic form:
[tex]\[0 = -x^2 + 3x - 5\][/tex]
Here, [tex]\(a = -1\)[/tex], [tex]\(b = 3\)[/tex], and [tex]\(c = -5\)[/tex]. These coefficients do not match the required values. Therefore, this equation does not meet the criteria.
4. [tex]\(0 = -3x + 5 - x^2\)[/tex]
Rearrange the terms to match the standard quadratic form:
[tex]\[0 = -x^2 - 3x + 5\][/tex]
Here, [tex]\(a = -1\)[/tex], [tex]\(b = -3\)[/tex], and [tex]\(c = 5\)[/tex]. These coefficients do not match the required values. Therefore, this equation does not meet the criteria.
Thus, the equation that has an [tex]\(a\)[/tex]-value of 1, a [tex]\(b\)[/tex]-value of -3, and a [tex]\(c\)[/tex]-value of -5 is:
[tex]\[
0 = -3x - 5 + x^2
\][/tex]
So, the correct equation is the first one presented:
[tex]\[
0 = -3x - 5 + x^2
\][/tex]
