how to determine to solve this
How many real solutions exist for this system of equations?
y = -x + 1
y = -x2 + 4x − 2
To determine the number of real solutions for this system of equations, we need to find the points of intersection between the two equations.
1. Set the equations equal to each other:
-x + 1 = -x^2 + 4x - 2
2. Rearrange the equation to have 0 on one side:
-x^2 + 4x - 2 + x - 1 = 0
-x^2 + 5x - 3 = 0
3. Determine the discriminant to see the number and nature of solutions:
The discriminant (D) can be found using the formula D = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation (ax^2 + bx + c = 0).
In this case, a = -1, b = 5, and c = -3. Plugging these values into the discriminant formula:
D = 5^2 - 4(-1)(-3)
= 25 - 12
= 13
4. Analyzing the discriminant:
If D > 0, there are two distinct real solutions.
If D = 0, there is one real solution.
If D < 0, there are no real solutions.
In this case, D = 13, which is greater than 0. Therefore, there are two distinct real solutions for the system of equations.
Note: To find the actual solutions, you can use the quadratic formula or factor the quadratic equation.