x^2+9x+2=0
Enter your answers, as exact values, in the boxes.
What is the solution of the equation when solved over the complex numbers?
x^2+24=0
The Correct Answer and Explanation is :
To solve the equations ( x^2 + 9x + 2 = 0 ) and ( x^2 + 24 = 0 ) over the complex numbers, we can use the quadratic formula and algebraic manipulations.
The correct answer is:
1. Solving ( x^2 + 9x + 2 = 0 )
The quadratic formula is given by:
[
x = \frac{{-b \pm \sqrt{{b^2 – 4ac}}}}{2a}
]
For our equation, ( a = 1 ), ( b = 9 ), and ( c = 2 ). Plugging in these values, we have:
[
x = \frac{{-9 \pm \sqrt{{9^2 – 4 \cdot 1 \cdot 2}}}}{2 \cdot 1}
]
Calculating the discriminant:
[
9^2 – 4 \cdot 1 \cdot 2 = 81 – 8 = 73
]
Now substituting back into the quadratic formula:
[
x = \frac{{-9 \pm \sqrt{73}}}{2}
]
So, the solutions to the equation ( x^2 + 9x + 2 = 0 ) are:
[
x = \frac{{-9 + \sqrt{73}}}{2} \quad \text{and} \quad x = \frac{{-9 – \sqrt{73}}}{2}
]
2. Solving ( x^2 + 24 = 0 )
For the second equation ( x^2 + 24 = 0 ), we can rearrange it to find ( x^2 = -24 ). Taking the square root of both sides gives:
[
x = \pm \sqrt{-24}
]
This can be simplified using imaginary numbers:
[
x = \pm \sqrt{24}i = \pm 2\sqrt{6}i
]
Summary of Solutions
- For the equation ( x^2 + 9x + 2 = 0 ):
- The solutions are ( x = \frac{{-9 + \sqrt{73}}}{2} ) and ( x = \frac{{-9 – \sqrt{73}}}{2} ).
- For the equation ( x^2 + 24 = 0 ):
- The solutions are ( x = 2\sqrt{6}i ) and ( x = -2\sqrt{6}i ).
These solutions demonstrate how the quadratic formula and the properties of imaginary numbers allow us to find solutions to equations that may not have real solutions. The results reflect the broader context of complex numbers in mathematics, providing a comprehensive solution set for polynomials.