A circle is defined by x² + y² = 25. A line given by y = x + 1 intersects the circle at two points, forming a chord. What is the…

Substitute $y = x + 1$ into $x^2 + y^2 = 25$: $x^2 + (x+1)^2 = 25$, which expands to $2x^2 + 2x + 1 = 25$, so $2x^2 + 2x - 24 = 0$ and $x^2 + x - 12 = 0$. This factors as $(x+4)(x-3) = 0$, so the endpoints of the chord have $x = 3$ and $x = -4$. The midpoint's $x$-coordinate is the average: $\frac{3 + (-4)}{2} = -\frac{1}{2}$. Shortcut: the sum of the roots is $-1$, so the average is $-\frac{1}{2}$. A common wrong answer is $-1$, which is the sum itself and was not halved.