Select The Correct Answer.What Are The Solutions Of This Quadratic Equation? X 2 + 2 X = − 2 X^2 + 2x = -2 X 2 + 2 X = − 2 A. X = 1 ± 1 X = 1 \pm 1 X = 1 ± 1 B. X = − 1 ± I X = -1 \pm I X = − 1 ± I C. X = 0 X = 0 X = 0 D. X = ± I X = \pm I X = ± I
Introduction
Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will explore the solutions of a quadratic equation and provide a step-by-step guide on how to solve it.
What is a Quadratic Equation?
A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two. The general form of a quadratic equation is:
ax^2 + bx + c = 0
where a, b, and c are constants, and x is the variable.
The Given Quadratic Equation
The given quadratic equation is:
x^2 + 2x = -2
To solve this equation, we need to rewrite it in the standard form of a quadratic equation, which is:
x^2 + 2x + 2 = 0
Step 1: Factorize the Quadratic Equation
The first step in solving a quadratic equation is to factorize it, if possible. Unfortunately, the given equation does not factorize easily, so we will need to use other methods to solve it.
Step 2: Use the Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations. It is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = 2, and c = 2. Plugging these values into the formula, we get:
x = (-(2) ± √((2)^2 - 4(1)(2))) / 2(1) x = (-2 ± √(4 - 8)) / 2 x = (-2 ± √(-4)) / 2
Step 3: Simplify the Expression
The expression under the square root is negative, which means that the quadratic equation has complex solutions. To simplify the expression, we can use the fact that √(-4) = 2i, where i is the imaginary unit.
x = (-2 ± 2i) / 2 x = -1 ± i
Conclusion
The solutions of the quadratic equation x^2 + 2x = -2 are x = -1 ± i. These solutions are complex numbers, which means that they have both real and imaginary parts.
Comparison with the Given Options
Let's compare our solutions with the given options:
A. x = 1 ± 1 B. x = -1 ± i C. x = 0 D. x = ± i
Our solutions match option B, which is x = -1 ± i.
Final Answer
The final answer is:
Introduction
Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will provide a Q&A guide on quadratic equations, covering common questions and topics.
Q: What is a quadratic equation?
A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two. The general form of a quadratic equation is:
ax^2 + bx + c = 0
where a, b, and c are constants, and x is the variable.
Q: How do I solve a quadratic equation?
A: There are several methods to solve a quadratic equation, including:
- Factoring: If the quadratic equation can be factored easily, you can solve it by finding the factors.
- Quadratic Formula: The quadratic formula is a powerful tool for solving quadratic equations. It is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
- Graphing: You can also solve a quadratic equation by graphing the related function and finding the x-intercepts.
Q: What is the quadratic formula?
A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
where a, b, and c are constants, and x is the variable.
Q: How do I use the quadratic formula?
A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula. For example, if you have the quadratic equation x^2 + 2x + 2 = 0, you can plug in a = 1, b = 2, and c = 2 into the formula.
Q: What are complex solutions?
A: Complex solutions are solutions that have both real and imaginary parts. They are often represented in the form a + bi, where a and b are real numbers, and i is the imaginary unit.
Q: How do I simplify complex solutions?
A: To simplify complex solutions, you can use the fact that i^2 = -1. For example, if you have the complex solution 2 + 3i, you can simplify it by multiplying the numerator and denominator by the conjugate of the denominator.
Q: What is the difference between real and complex solutions?
A: Real solutions are solutions that have only real parts, while complex solutions have both real and imaginary parts. Real solutions are often easier to work with, but complex solutions can be more interesting and challenging.
Q: Can I use a calculator to solve quadratic equations?
A: Yes, you can use a calculator to solve quadratic equations. Many calculators have built-in functions for solving quadratic equations, such as the quadratic formula.
Q: Are quadratic equations used in real-life applications?
A: Yes, quadratic equations are used in many real-life applications, such as:
- Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
- Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
- Economics: Quadratic equations are used to model economic systems and make predictions about future trends.
Conclusion
Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we have provided a Q&A guide on quadratic equations, covering common questions and topics. We hope that this guide has been helpful in understanding quadratic equations and their applications.