Use the substitution method to determine whether the linear-quadratic system y = x^2 - 2x - 6,
y= -5x - 9 has 0, 1, or 2 solutions.
To determine the number of solutions for the system, we need to find the value(s) of x that satisfy both equations.
First, we solve the second equation for y in terms of x:
y = -5x - 9
Now, we substitute this expression for y in the first equation:
x^2 - 2x - 6 = -5x - 9
Next, we simplify this equation:
x^2 - 2x - 6 + 5x + 9 = 0
x^2 + 3x + 3 = 0
This is a quadratic equation, and we can determine the number of solutions by considering the discriminant. The discriminant, Δ, is given by the formula: Δ = b^2 - 4ac. For our equation, a = 1, b = 3, and c = 3.
Δ = (3)^2 - 4(1)(3) = 9 - 12 = -3
Since the discriminant is negative (Δ < 0), the quadratic equation has no real solutions. Therefore, the system has no solutions.