Enter The Correct Answer In The Box.What Are The Solutions Of This Quadratic Equation? ${ X^2 + 164 = 16x }$Substitute The Values Of { A $}$ And { B $}$ To Complete The Solutions. ${ X = A - Bi }$
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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 delve into the world of quadratic equations and explore the solutions to a specific equation. We will also discuss the importance of quadratic equations in real-world applications and provide a step-by-step guide on how to solve them.
What are Quadratic Equations?
A quadratic equation is a polynomial equation of degree two, which means that 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 can be used to find the solutions to any quadratic equation, regardless of whether it can be factored or not.
Solving the Given Quadratic Equation
The given quadratic equation is:
x^2 + 164 = 16x
To solve this equation, we need to rewrite it in the standard form of a quadratic equation, which is:
ax^2 + bx + c = 0
Subtracting 16x from both sides of the equation, we get:
x^2 - 16x + 164 = 0
Now, we can see that the coefficients of the equation are a = 1, b = -16, and c = 164.
Substituting Values into the Quadratic Formula
To find the solutions to the equation, we can substitute the values of a, b, and c into the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Plugging in the values, we get:
x = (16 ± √((-16)^2 - 4(1)(164))) / 2(1)
Simplifying the expression, we get:
x = (16 ± √(256 - 656)) / 2
x = (16 ± √(-400)) / 2
x = (16 ± 20i) / 2
x = 8 ± 10i
Interpreting the Solutions
The solutions to the quadratic equation are x = 8 + 10i and x = 8 - 10i. These solutions are complex numbers, which means that they have both real and imaginary parts.
Real-World Applications of Quadratic Equations
Quadratic equations have numerous real-world applications, including:
- Physics: Quadratic equations are used to describe the motion of objects under the influence of gravity, friction, and other forces.
- Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
- Computer Science: Quadratic equations are used in algorithms for solving problems, such as finding the shortest path between two points in a graph.
Conclusion
In conclusion, solving quadratic equations is a crucial skill for students and professionals alike. The quadratic formula is a powerful tool for solving quadratic equations, and it can be used to find the solutions to any quadratic equation, regardless of whether it can be factored or not. We have seen how to solve a specific quadratic equation using the quadratic formula, and we have discussed the importance of quadratic equations in real-world applications.
Final Thoughts
Quadratic equations are a fundamental concept in mathematics, and they have numerous real-world applications. By understanding how to solve quadratic equations, we can gain a deeper appreciation for the beauty and power of mathematics. Whether you are a student or a professional, solving quadratic equations is an essential skill that can help you succeed in your field.
Additional Resources
- Quadratic Formula Calculator: A online calculator that can be used to solve quadratic equations.
- Quadratic Equation Solver: A software tool that can be used to solve quadratic equations.
- Mathematics Textbooks: A list of recommended textbooks for learning mathematics, including quadratic equations.
Frequently Asked Questions
- What is a quadratic equation? A quadratic equation is a polynomial equation of degree two, which means that the highest power of the variable (usually x) is two.
- How do I solve a quadratic equation? You can solve a quadratic equation using various methods, including factoring, completing the square, and the quadratic formula.
- What is the quadratic formula? The quadratic formula is a powerful tool for solving quadratic equations. It is given by: x = (-b ± √(b^2 - 4ac)) / 2a
Glossary of Terms
- Quadratic Equation: A polynomial equation of degree two, which means that the highest power of the variable (usually x) is two.
- Quadratic Formula: A powerful tool for solving quadratic equations. It is given by: x = (-b ± √(b^2 - 4ac)) / 2a
- Complex Number: A number that has both real and imaginary parts.
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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 answer some of the most frequently asked questions about quadratic equations, including how to solve them, what the quadratic formula is, and what complex numbers are.
Q: What is a quadratic equation?
A: A quadratic equation is a polynomial equation of degree two, which means that 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, completing the square, and the quadratic formula. The quadratic formula is a powerful tool for solving quadratic equations, and it is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
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 a complex number?
A: A complex number is a number that has both real and imaginary parts. It is typically written in the form:
a + bi
where a is the real part and b is the imaginary part.
Q: How do I simplify a complex number?
A: To simplify a complex number, you can use the following steps:
- Multiply the numerator and denominator by the conjugate of the denominator.
- Simplify the expression.
Q: What is the difference between a real number and a complex number?
A: A real number is a number that has no imaginary part, while a complex number is a number that has both real and imaginary parts.
Q: Can I use the quadratic formula to solve a quadratic equation with complex solutions?
A: Yes, you can use the quadratic formula to solve a quadratic equation with complex solutions. The quadratic formula will give you the complex solutions to the equation.
Q: How do I determine if a quadratic equation has real or complex solutions?
A: To determine if a quadratic equation has real or complex solutions, you can use the discriminant, which is given by:
b^2 - 4ac
If the discriminant is positive, the equation has two real solutions. If the discriminant is negative, the equation has two complex solutions.
Q: What is the discriminant?
A: The discriminant is a value that is used to determine if a quadratic equation has real or complex solutions. It is given by:
b^2 - 4ac
Q: Can I use the quadratic formula to solve a quadratic equation with a negative discriminant?
A: Yes, you can use the quadratic formula to solve a quadratic equation with a negative discriminant. The quadratic formula will give you the complex solutions to the equation.
Q: How do I find the solutions to a quadratic equation with a negative discriminant?
A: To find the solutions to a quadratic equation with a negative discriminant, you can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
The solutions will be complex numbers.
Q: What is the difference between a quadratic equation with real solutions and a quadratic equation with complex solutions?
A: A quadratic equation with real solutions has two real solutions, while a quadratic equation with complex solutions has two complex solutions.
Q: Can I use the quadratic formula to solve a quadratic equation with real solutions?
A: Yes, you can use the quadratic formula to solve a quadratic equation with real solutions. The quadratic formula will give you the real solutions to the equation.
Q: How do I determine if a quadratic equation has real or complex solutions using the quadratic formula?
A: To determine if a quadratic equation has real or complex solutions using the quadratic formula, you can look at the discriminant, which is given by:
b^2 - 4ac
If the discriminant is positive, the equation has two real solutions. If the discriminant is negative, the equation has two complex solutions.
Q: What is the significance of the quadratic formula in mathematics?
A: The quadratic formula is a powerful tool for solving quadratic equations, and it has numerous applications in mathematics, science, and engineering.
Q: Can I use the quadratic formula to solve a quadratic equation with a zero discriminant?
A: Yes, you can use the quadratic formula to solve a quadratic equation with a zero discriminant. The quadratic formula will give you one real solution or two real solutions, depending on the equation.
Q: How do I find the solutions to a quadratic equation with a zero discriminant?
A: To find the solutions to a quadratic equation with a zero discriminant, you can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
The solutions will be real numbers.
Q: What is the difference between a quadratic equation with a zero discriminant and a quadratic equation with a negative discriminant?
A: A quadratic equation with a zero discriminant has one real solution or two real solutions, while a quadratic equation with a negative discriminant has two complex solutions.
Q: Can I use the quadratic formula to solve a quadratic equation with a zero or negative discriminant?
A: Yes, you can use the quadratic formula to solve a quadratic equation with a zero or negative discriminant. The quadratic formula will give you the real or complex solutions to the equation.
Q: How do I determine if a quadratic equation has a zero or negative discriminant?
A: To determine if a quadratic equation has a zero or negative discriminant, you can look at the discriminant, which is given by:
b^2 - 4ac
If the discriminant is zero, the equation has one real solution or two real solutions. If the discriminant is negative, the equation has two complex solutions.
Q: What is the significance of the discriminant in mathematics?
A: The discriminant is a value that is used to determine if a quadratic equation has real or complex solutions. It is given by:
b^2 - 4ac
Q: Can I use the discriminant to determine if a quadratic equation has real or complex solutions?
A: Yes, you can use the discriminant to determine if a quadratic equation has real or complex solutions. If the discriminant is positive, the equation has two real solutions. If the discriminant is negative, the equation has two complex solutions.
Q: How do I find the discriminant of a quadratic equation?
A: To find the discriminant of a quadratic equation, you can use the formula:
b^2 - 4ac
Q: What is the difference between a quadratic equation with a positive discriminant and a quadratic equation with a negative discriminant?
A: A quadratic equation with a positive discriminant has two real solutions, while a quadratic equation with a negative discriminant has two complex solutions.
Q: Can I use the quadratic formula to solve a quadratic equation with a positive discriminant?
A: Yes, you can use the quadratic formula to solve a quadratic equation with a positive discriminant. The quadratic formula will give you the real solutions to the equation.
Q: How do I find the solutions to a quadratic equation with a positive discriminant?
A: To find the solutions to a quadratic equation with a positive discriminant, you can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
The solutions will be real numbers.
Q: What is the significance of the quadratic formula in mathematics and science?
A: The quadratic formula is a powerful tool for solving quadratic equations, and it has numerous applications in mathematics, science, and engineering.
Q: Can I use the quadratic formula to solve a quadratic equation with a zero or positive discriminant?
A: Yes, you can use the quadratic formula to solve a quadratic equation with a zero or positive discriminant. The quadratic formula will give you the real or complex solutions to the equation.
Q: How do I determine if a quadratic equation has a zero or positive discriminant?
A: To determine if a quadratic equation has a zero or positive discriminant, you can look at the discriminant, which is given by:
b^2 - 4ac
If the discriminant is zero, the equation has one real solution or two real solutions. If the discriminant is positive, the equation has two real solutions.
Q: What is the difference between a quadratic equation with a zero discriminant and a quadratic equation with a positive discriminant?
A: A quadratic equation with a zero discriminant has one real solution or two real solutions, while a quadratic equation with a positive discriminant has two real solutions.
Q: Can I use the quadratic formula to solve a quadratic equation with a zero or positive discriminant?
A: Yes, you can use the quadratic