What are the two solutions of 2x^2 = -X^2 - 5x - 1? A. the y-coordinates of the intersection points of the graphs of y = 2x^2 and y = x^2 – 5x - 1
B. the x-coordinates of the x-intercepts of the graphs of y = 2x^2 and y = -x^2 – 5x - 1
C. the x-coordinates of the intersection points of the graphs of y = 2x^2 and y = -x^2 – 5x - 1
D. the y-coordinates of the y-intercepts of the graphs of y = 2x^2 and y = -x^2–5x - 1

Step-by-step explanation:

Let's solve 2x^2 = -X^2 - 5x - 1.  Consolidate all terms on the left side and write 0 on the right side:

3x^2 + 5x + 1 = 0.  This is a quadratic equation.  Let's solve it for x using the quadratic formula:

a = 3, b = 5, c = 1, and so the discriminant is b^2 - 4ac = 5^2 - 4(3)(1) = 13.  Because the discriminant is positive, we know that there are two distinct, real roots; the graphs of y = 2x^2 and y = x^2 - 5x - 1 intersect in two places whose x-coordinates are the real roots mentioned above.

Answer A is not correct as stated, but would be correct if we were to replace "the y-coordinates" with "the x-coordinates."  

Answer C would be correct if and only if we write y = x^2 - 5x - 1.