How Many And What Type Of Solutions Does $6x^2 - 2x + 7 = 0$ Have?A. 1 Rational Solution B. 2 Rational Solutions C. 2 Nonreal Solutions D. 2 Irrational Solutions
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Introduction
Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable 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. In this article, we will focus on solving the quadratic equation 6x^2 - 2x + 7 = 0 and determine the number and type of solutions it has.
Understanding the Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations. It is given by:
x = (-b ± â(b^2 - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation. To use the quadratic formula, we need to identify the values of a, b, and c in the given equation.
Identifying the Coefficients
In the equation 6x^2 - 2x + 7 = 0, we can identify the coefficients as follows:
a = 6 (coefficient of x^2) b = -2 (coefficient of x) c = 7 (constant term)
Applying the Quadratic Formula
Now that we have identified the coefficients, we can apply the quadratic formula to solve the equation. Plugging in the values of a, b, and c, we get:
x = (2 ± â((-2)^2 - 4(6)(7))) / 2(6) x = (2 ± â(4 - 168)) / 12 x = (2 ± â(-164)) / 12
Analyzing the Solutions
The quadratic formula gives us two possible solutions for x. However, we need to analyze these solutions to determine their nature. The expression under the square root, -164, is negative, which means that the solutions will be complex numbers.
Complex Solutions
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. In our case, the solutions are:
x = (2 ± â(-164)) / 12 x = (2 ± iâ164) / 12
Conclusion
In conclusion, the quadratic equation 6x^2 - 2x + 7 = 0 has two complex solutions. These solutions are not rational, as they contain the imaginary unit i. Therefore, the correct answer is:
D. 2 nonreal solutions
Final Thoughts
Solving quadratic equations can be a challenging task, but with the right tools and techniques, it can be done efficiently. The quadratic formula is a powerful tool that can be used to solve quadratic equations, but it is essential to analyze the solutions to determine their nature. In this article, we have seen how to apply the quadratic formula to solve the equation 6x^2 - 2x + 7 = 0 and determine the number and type of solutions it has.
Additional Resources
For those who want to learn more about quadratic equations and the quadratic formula, here are some additional resources:
- Khan Academy: Quadratic Equations
- Mathway: Quadratic Formula
- Wolfram Alpha: Quadratic Formula
FAQs
Q: What is the quadratic formula?
A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by:
x = (-b ± â(b^2 - 4ac)) / 2a
Q: How do I apply the quadratic formula?
A: To apply the quadratic formula, you need to identify the values of a, b, and c in the given equation and plug them into the formula.
Q: What are complex solutions?
A: Complex solutions are numbers 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: How do I determine the nature of the solutions?
A: To determine the nature of the solutions, you need to analyze the expression under the square root. If it is negative, the solutions will be complex numbers.
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Introduction
Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In our previous article, we discussed how to solve the quadratic equation 6x^2 - 2x + 7 = 0 and determine the number and type of solutions it has. In this article, we will provide a comprehensive guide to quadratic equations, including a Q&A section to address common questions and concerns.
Understanding Quadratic Equations
A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable 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.
Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations. It is given by:
x = (-b ± â(b^2 - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation.
Q&A Section
Q: What is the quadratic formula?
A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by:
x = (-b ± â(b^2 - 4ac)) / 2a
Q: How do I apply the quadratic formula?
A: To apply the quadratic formula, you need to identify the values of a, b, and c in the given equation and plug them into the formula.
Q: What are complex solutions?
A: Complex solutions are numbers 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: How do I determine the nature of the solutions?
A: To determine the nature of the solutions, you need to analyze the expression under the square root. If it is negative, the solutions will be complex numbers.
Q: What is the difference between rational and irrational solutions?
A: Rational solutions are numbers that can be expressed as a fraction of two integers, while irrational solutions are numbers that cannot be expressed as a fraction of two integers.
Q: How do I know if a quadratic equation has real or complex solutions?
A: To determine if a quadratic equation has real or complex solutions, you need to analyze the expression under the square root. If it is non-negative, the solutions will be real numbers. If it is negative, the solutions will be complex numbers.
Q: Can a quadratic equation have more than two solutions?
A: No, a quadratic equation can have at most two solutions.
Q: How do I graph a quadratic equation?
A: To graph a quadratic equation, you need to identify the vertex of the parabola, which is the point where the parabola changes direction. You can use the quadratic formula to find the x-coordinate of the vertex.
Q: What is the significance of the discriminant in a quadratic equation?
A: The discriminant is the expression under the square root in the quadratic formula. It determines the nature of the solutions, whether they are real or complex.
Conclusion
In conclusion, quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. The quadratic formula is a powerful tool for solving quadratic equations, and it is essential to understand the nature of the solutions to determine their significance. We hope that this Q&A section has addressed common questions and concerns and provided a comprehensive guide to quadratic equations.
Additional Resources
For those who want to learn more about quadratic equations and the quadratic formula, here are some additional resources:
- Khan Academy: Quadratic Equations
- Mathway: Quadratic Formula
- Wolfram Alpha: Quadratic Formula
Final Thoughts
Solving quadratic equations can be a challenging task, but with the right tools and techniques, it can be done efficiently. The quadratic formula is a powerful tool that can be used to solve quadratic equations, but it is essential to analyze the solutions to determine their nature. We hope that this article has provided a comprehensive guide to quadratic equations and has addressed common questions and concerns.