Select All The Correct Equations.Which Equations Have No Real Solution But Have Two Complex Solutions?$\[ \begin{array}{|c|c|} \hline 3x^2 - 5x = -8 & 2x^2 = 6x - 5 \\ \hline 12x = 9x^2 + 4 & -x^2 - 10x = 34 \\ \hline \end{array} \\]
Introduction
In mathematics, equations can be classified into different types based on their solutions. Real solutions are the values of the variable that satisfy the equation and are real numbers. Complex solutions, on the other hand, are the values of the variable that satisfy the equation and are complex numbers. In this article, we will focus on selecting equations that have no real solution but have two complex solutions.
Understanding Complex Solutions
Complex solutions are the values of the variable that satisfy the equation and are complex numbers. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which satisfies the equation i^2 = -1. Complex solutions can be expressed in the form of a + bi, where a and b are real numbers.
Analyzing the Given Equations
The given equations are:
| Equation | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| 3x^2 - 5x = -8 | 2x^2 = 6x - 5 | 12x = 9x^2 + 4 | -x^2 - 10x = 34 |
We need to analyze each equation to determine if it has no real solution but has two complex solutions.
Equation 1: 3x^2 - 5x = -8
To analyze this equation, we need to rewrite it in the standard form of a quadratic equation, which is ax^2 + bx + c = 0. We can rewrite the equation as:
3x^2 - 5x + 8 = 0
The discriminant of this equation is b^2 - 4ac, which is (-5)^2 - 4(3)(8) = 25 - 96 = -71. Since the discriminant is negative, the equation has no real solution but has two complex solutions.
Equation 2: 2x^2 = 6x - 5
To analyze this equation, we need to rewrite it in the standard form of a quadratic equation, which is ax^2 + bx + c = 0. We can rewrite the equation as:
2x^2 - 6x + 5 = 0
The discriminant of this equation is b^2 - 4ac, which is (-6)^2 - 4(2)(5) = 36 - 40 = -4. Since the discriminant is negative, the equation has no real solution but has two complex solutions.
Equation 3: 12x = 9x^2 + 4
To analyze this equation, we need to rewrite it in the standard form of a quadratic equation, which is ax^2 + bx + c = 0. We can rewrite the equation as:
9x^2 - 12x + 4 = 0
The discriminant of this equation is b^2 - 4ac, which is (-12)^2 - 4(9)(4) = 144 - 144 = 0. Since the discriminant is zero, the equation has one real solution and one complex solution.
Equation 4: -x^2 - 10x = 34
To analyze this equation, we need to rewrite it in the standard form of a quadratic equation, which is ax^2 + bx + c = 0. We can rewrite the equation as:
-x^2 - 10x - 34 = 0
The discriminant of this equation is b^2 - 4ac, which is (-10)^2 - 4(-1)(-34) = 100 - 136 = -36. Since the discriminant is negative, the equation has no real solution but has two complex solutions.
Conclusion
In conclusion, the equations that have no real solution but have two complex solutions are:
- 3x^2 - 5x = -8
- 2x^2 = 6x - 5
- -x^2 - 10x = 34
These equations have a negative discriminant, which means they have no real solution but have two complex solutions.
References
- [1] "Quadratic Equations" by Math Open Reference
- [2] "Complex Numbers" by Math Is Fun
- [3] "Discriminant" by Wolfram MathWorld
Frequently Asked Questions
- Q: What is a complex solution? A: A complex solution is a value of the variable that satisfies the equation and is a complex number.
- Q: What is the discriminant of a quadratic equation? A: The discriminant of a quadratic equation is b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation.
- Q: How do I determine if an equation has no real solution but has two complex solutions?
A: To determine if an equation has no real solution but has two complex solutions, you need to calculate the discriminant of the equation. If the discriminant is negative, the equation has no real solution but has two complex solutions.
Frequently Asked Questions: Complex Solutions =============================================
Introduction
In our previous article, we discussed how to select equations that have no real solution but have two complex solutions. In this article, we will answer some frequently asked questions related to complex solutions.
Q&A
Q: What is a complex solution?
A: A complex solution is a value of the variable that satisfies the equation and is a complex number. A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, which satisfies the equation i^2 = -1.
Q: What is the discriminant of a quadratic equation?
A: The discriminant of a quadratic equation is b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation. The discriminant is used to determine the nature of the solutions of a quadratic equation.
Q: How do I determine if an equation has no real solution but has two complex solutions?
A: To determine if an equation has no real solution but has two complex solutions, you need to calculate the discriminant of the equation. If the discriminant is negative, the equation has no real solution but has two complex solutions.
Q: What is the difference between a real solution and a complex solution?
A: A real solution is a value of the variable that satisfies the equation and is a real number. A complex solution, on the other hand, is a value of the variable that satisfies the equation and is a complex number.
Q: Can a quadratic equation have both real and complex solutions?
A: Yes, a quadratic equation can have both real and complex solutions. This occurs when the discriminant of the equation is zero.
Q: How do I find the complex solutions of a quadratic equation?
A: To find the complex solutions of a quadratic equation, you need to use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. If the discriminant is negative, the solutions will be complex numbers.
Q: What is the imaginary unit?
A: The imaginary unit is a mathematical concept that is used to extend the real numbers to the complex numbers. It is denoted by the symbol i and satisfies the equation i^2 = -1.
Q: Can complex solutions be expressed in the form of a + bi?
A: Yes, complex solutions can be expressed in the form of a + bi, where a and b are real numbers and i is the imaginary unit.
Q: Are complex solutions always in the form of a + bi?
A: No, complex solutions are not always in the form of a + bi. They can also be expressed in the form of a + ki, where k is a real number.
Conclusion
In conclusion, complex solutions are an important concept in mathematics that can be used to solve quadratic equations. By understanding the discriminant and the quadratic formula, you can determine if an equation has no real solution but has two complex solutions.
References
- [1] "Quadratic Equations" by Math Open Reference
- [2] "Complex Numbers" by Math Is Fun
- [3] "Discriminant" by Wolfram MathWorld
Additional Resources
- [1] "Complex Solutions" by Khan Academy
- [2] "Quadratic Equations" by MIT OpenCourseWare
- [3] "Complex Numbers" by Wolfram Alpha