To find the discriminant of the given quadratic equation, we need to follow several steps. We'll start by rearranging the equation into the standard quadratic form [tex]\(ax^2 + bx + c = 0\)[/tex].
1. The original equation is:
[tex]\[ x^2 + 11x + 121 = x + 96 \][/tex]
2. Subtract [tex]\(x\)[/tex] and 96 from both sides to bring all terms to one side of the equation:
[tex]\[ x^2 + 11x + 121 - x - 96 = 0 \][/tex]
3. Simplify the equation by combining like terms:
[tex]\[ x^2 + (11x - x) + (121 - 96) = 0 \][/tex]
[tex]\[ x^2 + 10x + 25 = 0 \][/tex]
Now, the quadratic equation is in standard form [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = 10\)[/tex], and [tex]\(c = 25\)[/tex].
4. The discriminant ([tex]\(\Delta\)[/tex]) of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
5. Substitute the values [tex]\(a = 1\)[/tex], [tex]\(b = 10\)[/tex], and [tex]\(c = 25\)[/tex] into the discriminant formula:
[tex]\[ \Delta = 10^2 - 4 \cdot 1 \cdot 25 \][/tex]
[tex]\[ \Delta = 100 - 100 \][/tex]
[tex]\[ \Delta = 0 \][/tex]
The discriminant of the quadratic equation [tex]\(x^2 + 11x + 121 = x + 96\)[/tex] is [tex]\(0\)[/tex].
Hence, the best answer is:
[tex]\[ \boxed{0} \][/tex]
