Match Each Equation With Its Number Of Unique Solutions.Equations:- $y = 3x^2 - 6x + 3$- $y = -2x^2 + 9x - 11$- $y = -x^2 - 4x + 7$\[ \begin{tabular}{|c|c|} \hline Two Real Solutions & One Real Solution \\ \hline & \\ &
Introduction
Quadratic equations are a fundamental concept in mathematics, and understanding their solutions is crucial for various applications in science, engineering, and other fields. In this article, we will explore the process of solving quadratic equations and match each equation with its number of unique solutions.
What are Quadratic Equations?
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. Quadratic equations can be solved using various methods, including factoring, completing the square, and the quadratic formula.
The Quadratic Formula
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 the coefficients of the quadratic equation. The quadratic formula provides two solutions for the equation, which can be real or complex.
Real and Complex Solutions
Real solutions are values of x that satisfy the equation and are real numbers. Complex solutions, on the other hand, are values of x that satisfy the equation but are complex numbers, meaning they have an imaginary part.
Equation 1: y = 3x^2 - 6x + 3
To determine the number of unique solutions for this equation, we need to examine its discriminant, which is given by:
b^2 - 4ac
In this case, a = 3, b = -6, and c = 3. Plugging these values into the discriminant formula, we get:
(-6)^2 - 4(3)(3) = 36 - 36 = 0
Since the discriminant is zero, the equation has only one real solution.
Equation 2: y = -2x^2 + 9x - 11
For this equation, a = -2, b = 9, and c = -11. The discriminant is:
(9)^2 - 4(-2)(-11) = 81 - 88 = -7
Since the discriminant is negative, the equation has no real solutions. However, it has two complex solutions.
Equation 3: y = -x^2 - 4x + 7
In this case, a = -1, b = -4, and c = 7. The discriminant is:
(-4)^2 - 4(-1)(7) = 16 + 28 = 44
Since the discriminant is positive, the equation has two real solutions.
Conclusion
In conclusion, we have matched each equation with its number of unique solutions. Equation 1 has one real solution, Equation 2 has no real solutions but two complex solutions, and Equation 3 has two real solutions.
Discussion
Quadratic equations are a fundamental concept in mathematics, and understanding their solutions is crucial for various applications in science, engineering, and other fields. The quadratic formula is a powerful tool for solving quadratic equations, and the discriminant is a key factor in determining the number of unique solutions.
References
- [1] "Quadratic Equations" by Math Open Reference
- [2] "The Quadratic Formula" by Khan Academy
- [3] "Discriminant" by Wolfram MathWorld
Frequently Asked Questions
- 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.
- Q: How do I solve a quadratic equation? A: You can solve a quadratic equation using various methods, including factoring, completing the square, and the quadratic formula.
- Q: What is the discriminant? A: The discriminant is a key factor in determining the number of unique solutions for a quadratic equation. It is given by b^2 - 4ac.
Glossary
- Quadratic equation: A polynomial equation of degree two.
- Discriminant: A key factor in determining the number of unique solutions for a quadratic equation.
- Real solution: A value of x that satisfies the equation and is a real number.
- Complex solution: A value of x that satisfies the equation but is a complex number.
Quadratic Equations: A Q&A Guide =====================================
Introduction
Quadratic equations are a fundamental concept in mathematics, and understanding their solutions is crucial for various applications in science, engineering, and other fields. In this article, we will provide a comprehensive Q&A guide to quadratic equations, covering topics such as solving quadratic equations, the quadratic formula, and the discriminant.
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: You can solve a quadratic equation using various methods, including:
- Factoring: If the quadratic expression can be factored into the product of two binomials, you can solve the equation by setting each factor equal to zero.
- Completing the square: This method involves rewriting the quadratic expression in a perfect square form, which allows you to solve the equation by setting the square equal to a constant.
- The quadratic formula: This is a powerful tool for solving quadratic equations, and it is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation.
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 the coefficients of the quadratic equation.
Q: What is the discriminant?
A: The discriminant is a key factor in determining the number of unique solutions for a quadratic equation. It is given by:
b^2 - 4ac
If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.
Q: How do I determine the number of unique solutions for a quadratic equation?
A: To determine the number of unique solutions for a quadratic equation, you need to examine the discriminant. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.
Q: What is the difference between a real solution and a complex solution?
A: A real solution is a value of x that satisfies the equation and is a real number. A complex solution, on the other hand, is a value of x that satisfies the equation but is a complex number, meaning it has an imaginary part.
Q: How do I graph a quadratic equation?
A: To graph a quadratic equation, you can use the following steps:
- Find the x-intercepts of the equation by setting the equation equal to zero and solving for x.
- Find the y-intercept of the equation by substituting x = 0 into the equation.
- Plot the x-intercepts and the y-intercept on a coordinate plane.
- Draw a smooth curve through the points to form the graph of the equation.
Q: What are some common applications of quadratic equations?
A: Quadratic equations have many applications in science, engineering, and other fields, including:
- 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 the behavior of economic systems, such as supply and demand curves.
Conclusion
In conclusion, quadratic equations are a fundamental concept in mathematics, and understanding their solutions is crucial for various applications in science, engineering, and other fields. This Q&A guide has provided a comprehensive overview of quadratic equations, covering topics such as solving quadratic equations, the quadratic formula, and the discriminant.
Frequently Asked Questions
- Q: What is a quadratic equation? A: A quadratic equation is a polynomial equation of degree two.
- Q: How do I solve a quadratic equation? A: You can solve a quadratic equation using various methods, including factoring, completing the square, and the quadratic formula.
- Q: What is the discriminant? A: The discriminant is a key factor in determining the number of unique solutions for a quadratic equation.
Glossary
- Quadratic equation: A polynomial equation of degree two.
- Discriminant: A key factor in determining the number of unique solutions for a quadratic equation.
- Real solution: A value of x that satisfies the equation and is a real number.
- Complex solution: A value of x that satisfies the equation but is a complex number.