Use the elimination method to solve the linear-quadratic system y = x^2 + 10, y= -7x -2.
To solve the system using the elimination method, we need to eliminate one variable by manipulating the equations.
Let's start by manipulating the second equation by multiplying it by x. This will allow us to eliminate the x variable:
x * y = x * (-7x - 2)
xy = -7x^2 - 2x
Now we can substitute this expression for xy in the first equation:
x^2 + 10 = -7x^2 - 2x
Rearrange the equation to get all terms on one side:
8x^2 + 2x + 10 = 0
To solve this quadratic equation, we can use 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 = 8, b = 2, and c = 10. Plugging these values into the quadratic formula, we get:
x = (-(2) ± √((2)^2 - 4(8)(10))) / 2(8)
Simplifying:
x = (-2 ± √(4 - 320)) / 16
x = (-2 ± √(-316)) / 16
Since the discriminant (√(b^2 - 4ac)) in the quadratic formula is negative, there are no real solutions for x.
Therefore, the linear-quadratic system has no solution.