Without performing construction, determine whether the parabola y = x ^ 2 and the straight line y = 12-x intersect.

If a parabola and a straight line intersect, that is, they have one or two common points, then at these points the values of y will coincide. Therefore, we equate the right-hand sides of the formulas of these functions. If the resulting equation has a solution, then the line and the parabola intersect.

x² = 12 – x

x² + x – 12 = 0

Coefficients: a = 1, b = 1, c = -12.

Let’s find the discriminant:

d = b² – 4ac = 1 – 4 * 1 * (-12) = 1 + 48 = 49> 0.

Since the discriminant is positive, the equation has two solutions, hence the parabola and the straight line have two intersection points.

Answer: the parabola y = x² and the straight line y = 12 – x intersect.