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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 explore the solutions of a quadratic equation and provide a step-by-step guide on how to complete the solutions.
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 - 6x = -58
To solve this equation, we need to rewrite it in the standard form of a quadratic equation, which is:
ax^2 + bx + c = 0
Rearranging the Equation
Let's rearrange the given equation to get:
x^2 - 6x + 58 = 0
Now, we have the equation in the standard form.
Identifying the Values of a, b, and c
In the standard form of a quadratic equation, a, b, and c are the coefficients of the terms. In this case:
a = 1 (coefficient of x^2) b = -6 (coefficient of x) c = 58 (constant term)
Completing the Solutions
To complete the solutions, we need to use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Substituting the values of a, b, and c, we get:
x = (6 ± √((-6)^2 - 4(1)(58))) / 2(1) x = (6 ± √(36 - 232)) / 2 x = (6 ± √(-196)) / 2
Simplifying the Expression
The expression inside the square root is negative, which means the solutions will be complex numbers. To simplify the expression, we can use the fact that √(-1) = i, where i is the imaginary unit.
x = (6 ± √(-196)) / 2 x = (6 ± 14i) / 2
Simplifying Further
We can simplify the expression further by dividing both the real and imaginary parts by 2:
x = 3 ± 7i
Conclusion
In this article, we solved a quadratic equation and completed the solutions using the quadratic formula. We identified the values of a, b, and c, and substituted them into the quadratic formula to get the solutions. The solutions are complex numbers, which is a common outcome when solving quadratic equations.
Tips and Tricks
- When solving quadratic equations, make sure to identify the values of a, b, and c correctly.
- Use the quadratic formula to complete the solutions.
- Simplify the expression inside the square root by using the fact that √(-1) = i.
- Be careful when dividing complex numbers, as the result may be different from what you expect.
Practice Problems
Try solving the following quadratic equations:
- x^2 + 5x = 14
- x^2 - 2x = 15
- x^2 + 3x = 12
Use the quadratic formula to complete the solutions and simplify the expressions.
References
- [1] "Quadratic Equations" by Math Open Reference
- [2] "Quadratic Formula" by Khan Academy
- [3] "Complex Numbers" by Wolfram MathWorld
Note: The references provided are for informational purposes only and are not intended to be a comprehensive list of resources.
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Introduction
Quadratic equations can be a challenging topic for many students and professionals. In this article, we will address some of the most frequently asked questions about quadratic equations, providing clear and concise answers to help you better understand this important concept.
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: To solve a quadratic equation, you can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Substitute the values of a, b, and c into the formula, and simplify the expression to get the solutions.
Q: What is the quadratic formula?
A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is:
x = (-b ± √(b^2 - 4ac)) / 2a
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. In other words, a quadratic equation has a highest power of two, while a linear equation has a highest power of one.
Q: Can a quadratic equation have more than two solutions?
A: No, a quadratic equation can have at most two solutions. This is because the quadratic formula provides two possible values for x, and these values are the only solutions to the equation.
Q: What is the significance of the discriminant (b^2 - 4ac) in the quadratic formula?
A: The discriminant is the expression inside the square root in the quadratic formula. If the discriminant is positive, the solutions will be real numbers. If the discriminant is zero, the solutions will be equal. If the discriminant is negative, the solutions will be complex numbers.
Q: Can a quadratic equation have complex solutions?
A: Yes, a quadratic equation can have complex solutions. This occurs when the discriminant is negative, and the solutions will be in the form of 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 √(-1) = i, where i is the imaginary unit. You can also use the conjugate of the complex number to simplify the expression.
Q: What is the conjugate of a complex number?
A: The conjugate of a complex number a + bi is a - bi.
Q: Can a quadratic equation have rational solutions?
A: Yes, a quadratic equation can have rational solutions. This occurs when the discriminant is a perfect square, and the solutions will be rational numbers.
Q: How do I determine if a quadratic equation has rational solutions?
A: To determine if a quadratic equation has rational solutions, you can use the discriminant. If the discriminant is a perfect square, the solutions will be rational numbers.
Conclusion
In this article, we addressed some of the most frequently asked questions about quadratic equations, providing clear and concise answers to help you better understand this important concept. Whether you are a student or a professional, quadratic equations are an essential part of mathematics, and understanding them is crucial for success in many fields.
Tips and Tricks
- Make sure to identify the values of a, b, and c correctly when solving a quadratic equation.
- Use the quadratic formula to complete the solutions.
- Simplify the expression inside the square root by using the fact that √(-1) = i.
- Be careful when dividing complex numbers, as the result may be different from what you expect.
Practice Problems
Try solving the following quadratic equations:
- x^2 + 5x = 14
- x^2 - 2x = 15
- x^2 + 3x = 12
Use the quadratic formula to complete the solutions and simplify the expressions.
References
- [1] "Quadratic Equations" by Math Open Reference
- [2] "Quadratic Formula" by Khan Academy
- [3] "Complex Numbers" by Wolfram MathWorld
Note: The references provided are for informational purposes only and are not intended to be a comprehensive list of resources.