Final answer:
The equation 0.75x(x+40)=0.35(x+20)+0.35(c+20) has two solutions.
Step-by-step explanation:
To solve the given equation 0.75x(x+40)=0.35(x+20)+0.35(c+20), we can first simplify both sides of the equation by performing the necessary algebraic operations. Distributing 0.75 on the left side and 0.35 on the right side gives us 0.75x^2 + 30x = 0.35x + 7 + 0.35c + 7.
Next, we can combine the like terms on both sides of the equation. Subtracting 0.35x and 14 from both sides gives us 0.75x^2 + 30x - 0.35x - 14 = 0.35c.
Simplifying further, we have 0.75x^2 + 29.65x - 14 = 0.35c.
This equation represents a quadratic equation. To find the number of solutions, we can examine the discriminant. The discriminant, given by b^2 - 4ac, tells us how many solutions exist. In this case, a = 0.75, b = 29.65, and c = -14. Substituting these values into the discriminant formula:
discriminant = (29.65)^2 - 4(0.75)(-14)
discriminant = 879.5225 + 42
discriminant = 921.5225
Since the discriminant is positive, there are two distinct solutions for the equation.