Question

Carlos graphs the equations y = 0.5x^2 + 3 and y = –4x^2 + 24x – 35 and generates the graph below. Which conclusion is supported by the graph?
A. The equation 0.5x2 + 3 = –4x2 + 24x – 35 has no solutions.
B. The equation 0.5x2 + 3 = –4x2 + 24x – 35 has one solution.
C. The equation 0.5x2 + 3 = –4x2 + 24x – 35 has two solutions.
D. The equation 0.5x2 + 3 = –4x2 + 24x – 35 has three solutions.

Answer 1

Answer: The correct option is A. The equation 0.5x2 + 3 = –4x2 + 24x – 35 has no solutions.

Explanation:

The given equations are,

$y=0.5x^2+3$

$y=-4x^2+24x-35$

Since the above equations are quadratic equations, therefore they area parabola.

The coefficient of first equation is a positive therefore it is an upward parabola. The coefficient of second equation is a negative therefore it is an downward parabola.

From given figure it is noticed that the vertex of first parabola is (0,3) and the vertex of second parabola is (3,1).

The vertex is the extreme point of the parabola. So for first equation $y\geq 3$ and for second equation $y\leq 1$. Therefore the parabolas will never intersect each other.

Since there is no intersection between parabolas therefore the equation

$0.5x^2+3=-4x^2+24x-35$ has no solution and the correct option is A.

Answer 2

 A is the correct answer because the parabolas never touch.