Please help me solve this question

Question

Study the system of equations.       

     Equation 1:   y=x2+5

     Equation 2:   y=2x+3

 

Part A. Graph the system of equations.

Part B. Prove algebraically that the system has no real solutions. Show all your work.

Part C. 

1. Apply a transformation to either Equation 1 or Equation 2 so that the system has at least one solution.
2. Justify why the change creates a solution to the system. 

Answer 1

x^2 +5 = 2x + 3

x^2 -2x +2 = 0

4-4(2) <0, the discriminate is negative. that means no real solutions, and 2 imaginary solutions

the first equation is a parabola upward opening with y intercept = 5

the 2nd equation is a straight line with slope = 2 and y intercept = 3. It lies completely below and to the right of the parabola, with no intersection. The parabola has vertex or minimum point at (0,5) which is also the y intercept.

shift the parabola down or shift the line up, either by 2 or more and you'll get 1 or 2 intersections, one or two solutions.

y=x^2 + 3 and y=2x+3 will have one solution (0,3)