Question

in the xy - plane, the graph of the equation $y = -x^2 + 9x - 100$ intersects the line $y = c$ at exactly one point. what is the value of $c$?
a) $-\frac{481}{4}$
b) $-100$
c) $-\frac{319}{4}$
d) $-\frac{9}{2}$

Explanation:

Step1: Set equations equal

Set $y = c$ equal to the quadratic:
$$c = -x^2 + 9x - 100$$

Step2: Rearrange to standard quadratic

Rearrange into $ax^2+bx+d=0$ form:
$$x^2 - 9x + (100 + c) = 0$$

Step3: Use discriminant for 1 solution

For one intersection, discriminant $\Delta = b^2-4ac=0$:
$$(-9)^2 - 4(1)(100 + c) = 0$$

Step4: Solve for c

Calculate and isolate $c$:
$$81 - 400 - 4c = 0$$
$$-319 - 4c = 0$$
$$c = -\frac{319}{4}$$

Answer:

C) $-\frac{319}{4}$