Let's examine each of the given equations to determine which one has an [tex]\( a \)[/tex]-value of 1, a [tex]\( b \)[/tex]-value of -3, and a [tex]\( c \)[/tex]-value of -5. These values correspond to the coefficients in a quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex].
1. The first equation is:
[tex]\[
0 = -3x - 5 + x^2
\][/tex]
Rearranging this, we get:
[tex]\[
x^2 - 3x - 5 = 0
\][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = -3 \)[/tex], and [tex]\( c = -5 \)[/tex].
2. The second equation is:
[tex]\[
0 = x - 3 - 5x^2
\][/tex]
Rearranging this, we get:
[tex]\[
-5x^2 + x - 3 = 0
\][/tex]
Here, [tex]\( a = -5 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -3 \)[/tex].
3. The third equation is:
[tex]\[
0 = 3x - 5 - x^2
\][/tex]
Rearranging this, we get:
[tex]\[
-x^2 + 3x - 5 = 0
\][/tex]
Here, [tex]\( a = -1 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = -5 \)[/tex].
4. The fourth equation is:
[tex]\[
0 = -3x + 5 - x^2
\][/tex]
Rearranging this, we get:
[tex]\[
-x^2 - 3x + 5 = 0
\][/tex]
Here, [tex]\( a = -1 \)[/tex], [tex]\( b = -3 \)[/tex], and [tex]\( c = 5 \)[/tex].
Comparing the coefficients, we see that the first equation matches the desired values:
[tex]\[
a = 1, b = -3, \text{ and } c = -5
\][/tex]
Therefore, the equation which 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]
The answer is the first equation.
