Two equations and their graphs are given. Find the intersection point(s) of the graphs by solving the system. x2  +  y  =  5x  −  2y  =  0(x, y) = ( ) (smaller x-value)(x, y) = ( ) (larger x-value)

The problem is asking to find the intersection points of the two given equations:

1. x^2 + y = 5
2. x - 2y = 0

Let's solve this system of equations step by step:

Step 1: Solve the second equation for x: x = 2y

Step 2: Substitute x = 2y into the first equation: (2y)^2 + y = 5 4y^2 + y - 5 = 0

Step 3: Solve the quadratic equation 4y^2 + y - 5 = 0. This can be done by factoring, completing the square, or using the quadratic formula. The quadratic formula is:

y = [-b ± sqrt(b^2 - 4ac)] / (2a)

In this case, a = 4, b = 1, and c = -5. Plugging these values into the quadratic formula gives:

y = [-1 ± sqrt((1)^2 - 44(-5))] / (2*4) y = [-1 ± sqrt(1 + 80)] / 8 y = [-1 ± sqrt(81)] / 8 y = [-1 ± 9] / 8

So the solutions for y are y = 1 and y = -10/8 = -1.25.

Step 4: Substitute y = 1 and y = -1.25 into the equation x = 2y to find the corresponding x values:

For y = 1, x = 21 = 2. For y = -1.25, x = 2(-1.25) = -2.5.

So the intersection points of the two graphs are (2, 1) and (-2.5, -1.25).