To solve the system of equations given by y = x² + 1 and y = 2x² – 4, we first need to find the points where the two graphs intersect. This can be done by setting the equations equal to each other:
x² + 1 = 2x² – 4
Rearranging this equation leads to:
0 = 2x² – x² – 4 – 1
This simplifies to:
x² – 5 = 0
We can now solve for x:
x² = 5
x = ±√5
Now, we can substitute these values back into either original equation to find the corresponding y coordinates. Let’s use the first equation:
For x = √5:
y = (√5)² + 1 = 5 + 1 = 6
For x = -√5:
y = (-√5)² + 1 = 5 + 1 = 6
Thus, the intersection points are (√5, 6) and (-√5, 6). The graphs of the equations will intersect at these points, and that’s where the solution to the system is found.
To visualize this, you would expect to see a standard parabola for both equations, with the first one opening upwards and the second one also opening upwards but steeper. The solution points would be visible on the same horizontal line at y = 6 where they intersect.