To determine the greatest possible value of [tex]\( b \)[/tex] for which the quadratic equation [tex]\( -x^2 + bx - 676 = 0 \)[/tex] has no real solutions, we need to analyze the discriminant of the quadratic equation.
For a quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex], the discriminant [tex]\( \Delta \)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
In our equation:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = b \)[/tex]
- [tex]\( c = -676 \)[/tex]
Substituting these values into the discriminant formula, we get:
[tex]\[
\Delta = b^2 - 4(-1)(-676)
\][/tex]
Simplifying within the parentheses:
[tex]\[
\Delta = b^2 - 4 \times 676
\][/tex]
Calculating [tex]\( 4 \times 676 \)[/tex] gives:
[tex]\[
4 \times 676 = 2704
\][/tex]
Thus, the discriminant simplifies to:
[tex]\[
\Delta = b^2 - 2704
\][/tex]
For the quadratic equation to have no real solutions, the discriminant must be less than zero:
[tex]\[
b^2 - 2704 < 0
\][/tex]
Solving this inequality for [tex]\( b \)[/tex]:
[tex]\[
b^2 < 2704
\][/tex]
Taking the square root of both sides, we obtain:
[tex]\[
|b| < \sqrt{2704}
\][/tex]
Since [tex]\( b \)[/tex] is a positive integer, we consider:
[tex]\[
b < \sqrt{2704}
\][/tex]
Calculating the square root of 2704 gives:
[tex]\[
\sqrt{2704} = 52
\][/tex]
Therefore, the greatest possible value of [tex]\( b \)[/tex] that satisfies this inequality is:
[tex]\[
b < 52
\][/tex]
Since [tex]\( b \)[/tex] must be an integer, the greatest possible value of [tex]\( b \)[/tex] is:
[tex]\[
\boxed{51}
\][/tex]
Thus, [tex]\( b \)[/tex] can be at most 51 for the quadratic equation [tex]\( -x^2 + bx - 676 = 0 \)[/tex] to have no real solutions.
