Which of the following represents all solutions (x,y) to the system of equations created by the linear equation and the quadratic equation y=x^2+9
(3,0) and (_3,0)
To find the solutions to the system of equations created by a linear equation and a quadratic equation, we need to solve the equations simultaneously.
Let's consider the linear equation y = x and the quadratic equation y = x^2 + 9.
Since both equations have y on one side, we can set them equal to each other:
x = x^2 + 9
Now, rearrange the equation to bring all terms to one side:
x^2 - x + 9 = 0
This is a quadratic equation, and to find its solutions, we can apply the quadratic formula. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 1, b = -1, and c = 9. Substituting these values into the quadratic formula, we get:
x = (-(-1) ± √((-1)^2 - 4(1)(9))) / (2(1))
x = (1 ± √(1 - 36)) / 2
x = (1 ± √(-35)) / 2
Since the discriminant (√(1 - 36)) is negative, we have a complex root. Therefore, there are no real solutions to the quadratic equation x^2 - x + 9 = 0.
As a result, there are no solutions (x, y) that satisfy both the linear equation y = x and the quadratic equation y = x^2 + 9 simultaneously.