One of the solutions to the equation x^3 + 5x^2 + 10x + 50 = 0 is -5. Find the full solution set. Which of the following options correctly describes the solution set?
It has one real solution and two non-real solutions.
It has one non-real solution and two solutions.
It has three real solutions.
huh? If the full solution set is {-5, -2√10, 2√10}, then there are three real solutions!
You are correct, I apologize for my mistake.
The full solution set {-5, -2√10, 2√10} does indeed contain three real solutions. Therefore, the correct option is: It has three real solutions.
To find the full solution set of a cubic equation, we can use the Rational Root Theorem or graphing techniques to identify any rational roots, and then apply the quadratic formula to find any remaining complex roots.
Given that one of the solutions to the equation x³ + 5x² + 10x + 50 = 0 is x = -5, we can use synthetic division or substitution to verify whether it is a real root. By dividing the cubic equation by (x + 5), we can see that the other factor is a quadratic equation.
Using synthetic division, the division of x³ + 5x² + 10x + 50 by (x + 5) yields (x² + 10). Since there is no remainder in the division, we can confirm that x = -5 is a real root.
Now we have a quadratic equation, x² + 10 = 0, which we can solve using the quadratic formula.
The quadratic formula is given by x = [-b ± √(b² - 4ac)] / (2a), where a, b, and c are the coefficients of the quadratic equation (ax² + bx + c = 0).
For the equation x² + 10 = 0, a = 1, b = 0, and c = 10. Plugging these values into the quadratic formula, we get:
x = [-0 ± √(0² - 4(1)(10))] / (2(1))
x = ±√(-40) / 2
Since the discriminant, the value inside the square root, is negative (-40 in this case), the quadratic equation does not have real solutions. Instead, it has two complex solutions.
Therefore, the solution set consists of one real solution (x = -5) and two non-real solutions. The correct option is: "It has one real solution and two non-real solutions."
The given equation is a cubic equation, and we are told that one of the solutions is -5. Therefore, we can use polynomial division or synthetic division to divide the equation by (x + 5).
Performing synthetic division, we have:
```
-5 | 1 5 10 50
| -5 0 -50
--------------
1 0 -40 0
```
The quotient is x^2 + 0x - 40, which can be factored as (x + 0)(x - 2√10)(x + 2√10).
Therefore, the full solution set is {-5, -2√10, 2√10}, which consists of one real solution (-5) and two non-real solutions (-2√10 and 2√10).
Therefore, the correct option is: It has one real solution and two non-real solutions.