How many real solutions does x² - 4x + 7 = 0 have?

The number of real solutions depends on the discriminant, $b^2-4ac$. Here $a=1$, $b=-4$, and $c=7$, so the discriminant is $16-28=-12$. A negative discriminant means no real solutions, choice F. Completing the square confirms it: $x^2-4x+7=(x-2)^2+3$, which is always at least $3$, so it is never zero. Choice G (one solution) would need a discriminant of $0$. Choice H (two solutions) would need a positive discriminant. Choice J is never possible for a quadratic equation. Graphically, the parabola lies entirely above the $x$-axis.