Enter The Correct Answer In The Box.What Are The Solutions Of This Quadratic Equation? X 2 − 6 X = − 58 X^2 - 6x = -58 X 2 − 6 X = − 58 Substitute The Values Of A A A And B B B To Complete The Solutions.
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Understanding Quadratic Equations
Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including 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.
The Given Quadratic Equation
In this article, we will focus on solving the quadratic equation:
x^2 - 6x = -58
To solve this equation, we need to rewrite it in the standard form of a quadratic equation, which is:
x^2 - 6x + 58 = 0
Substituting Values of a and b
Now that we have the quadratic equation in the standard form, we can identify the values of a and b. In this case, a = 1 and b = -6.
Using 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.
Applying the Quadratic Formula
Now that we have the values of a and b, we can substitute them into the quadratic formula:
x = (6 ± √((-6)^2 - 4(1)(58))) / 2(1)
Simplifying the expression under the square root, we get:
x = (6 ± √(36 - 232)) / 2
x = (6 ± √(-196)) / 2
Simplifying the Expression
The expression under the square root is negative, which means that the quadratic equation has no real solutions. However, we can simplify the expression by using complex numbers.
x = (6 ± √(-196)) / 2
x = (6 ± 14i) / 2
x = 3 ± 7i
Conclusion
In this article, we have solved the quadratic equation x^2 - 6x = -58 using the quadratic formula. We have identified the values of a and b, substituted them into the quadratic formula, and simplified the expression to obtain the solutions.
The solutions of the quadratic equation are:
- x = 3 + 7i
- x = 3 - 7i
These solutions are complex numbers, which means that the quadratic equation has no real solutions.
Importance of Quadratic Equations
Quadratic equations are an essential part of mathematics, and they have numerous applications in various fields. They are used to model real-world problems, such as the motion of objects, the growth of populations, and the behavior of electrical circuits.
Real-World Applications of Quadratic Equations
Quadratic equations have numerous real-world applications, including:
- Physics: Quadratic equations are used to model the motion of objects, such as the trajectory of a projectile or the motion of a pendulum.
- Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electrical circuits.
- Economics: Quadratic equations are used to model the behavior of economic systems, such as the growth of populations and the behavior of financial markets.
Tips for Solving Quadratic Equations
Solving quadratic equations can be challenging, but there are several tips that can help:
- Use the quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations.
- Simplify the expression: Simplifying the expression under the square root can make it easier to solve the equation.
- Use complex numbers: Complex numbers can be used to solve quadratic equations that have no real solutions.
Conclusion
In this article, we have solved the quadratic equation x^2 - 6x = -58 using the quadratic formula. We have identified the values of a and b, substituted them into the quadratic formula, and simplified the expression to obtain the solutions. We have also discussed the importance of quadratic equations and their real-world applications.
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What is a Quadratic Equation?
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.
Q: What are the solutions of a quadratic equation?
A: The solutions of a quadratic equation are the values of x that satisfy the equation. There are three possible cases:
- Two distinct real solutions: In this case, the quadratic equation has two distinct real solutions.
- One repeated real solution: In this case, the quadratic equation has one repeated real solution.
- No real solutions: In this case, the quadratic equation has no real solutions, but it may have complex solutions.
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, 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
where a, b, and c are the coefficients of the quadratic equation.
- 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 the coefficients of the quadratic equation.
Q: How do I use the quadratic formula?
A: To use the quadratic formula, you need to:
- Identify the coefficients: Identify the coefficients a, b, and c of the quadratic equation.
- Plug in the values: Plug in the values of a, b, and c into the quadratic formula.
- Simplify the expression: Simplify the expression under the square root.
- Find the solutions: Find the solutions of the quadratic equation.
Q: What is the difference between a quadratic equation and a linear equation?
A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. The general form of a linear equation is:
ax + b = 0
where a and b are constants, and x is the variable.
Q: Can a quadratic equation have no real solutions?
A: Yes, a quadratic equation can have no real solutions. This occurs when the expression under the square root is negative.
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 of the quadratic equation. If the discriminant is positive, the quadratic equation has two distinct real solutions. If the discriminant is zero, the quadratic equation has one repeated real solution. If the discriminant is negative, the quadratic equation has no real solutions.
Q: Can a quadratic equation have complex solutions?
A: Yes, a quadratic equation can have complex solutions. This occurs when the expression under the square root is negative.
Q: How do I graph a quadratic equation?
A: To graph a quadratic equation, you need to:
- Plot the related function: Plot the related function on a coordinate plane.
- Find the x-intercepts: Find the x-intercepts of the related function.
- Draw the graph: Draw the graph of the quadratic equation.
Q: What is the vertex of a quadratic equation?
A: The vertex of a quadratic equation is the point on the graph where the quadratic equation reaches its maximum or minimum value.
Q: How do I find the vertex of a quadratic equation?
A: To find the vertex of a quadratic equation, you need to:
- Use the formula: Use the formula x = -b / 2a to find the x-coordinate of the vertex.
- Find the y-coordinate: Find the y-coordinate of the vertex by plugging the x-coordinate into the quadratic equation.
Q: What is the axis of symmetry of a quadratic equation?
A: The axis of symmetry of a quadratic equation is the vertical line that passes through the vertex of the quadratic equation.
Q: How do I find the axis of symmetry of a quadratic equation?
A: To find the axis of symmetry of a quadratic equation, you need to:
- Use the formula: Use the formula x = -b / 2a to find the x-coordinate of the axis of symmetry.
- Draw the line: Draw the vertical line that passes through the x-coordinate.
Conclusion
In this article, we have answered some of the most frequently asked questions about quadratic equations. We have discussed the definition of a quadratic equation, the solutions of a quadratic equation, and the methods for solving a quadratic equation. We have also discussed the significance of the discriminant, the vertex, and the axis of symmetry of a quadratic equation.