Choose The Answer Below That:1. Shows The Following Quadratic Equation Correctly Rewritten Using The Quadratic Formula.2. Lists The Correct Solutions.Given Quadratic Equation:${ 4x^2 = -8x + 221 }$Options:A. $[ X = \frac{8 \pm
Understanding Quadratic Equations
A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It has the general form of ax^2 + bx + c = 0, where a, b, and c are constants. 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 of a quadratic equation, even if it cannot be factored easily.
Given Quadratic Equation
The given quadratic equation is:
4x^2 = -8x + 221
To rewrite this equation using the quadratic formula, we need to rewrite it in the standard form of ax^2 + bx + c = 0. We can do this by subtracting 221 from both sides of the equation and then dividing both sides by 4.
4x^2 + 8x - 221 = 0
Now we have the quadratic equation in the standard form. We can use the quadratic formula to find the solutions.
Rewriting the Quadratic Equation Using the Quadratic Formula
To rewrite the quadratic equation using the quadratic formula, we need to identify the values of a, b, and c. In this case, a = 4, b = 8, and c = -221.
x = (-b ± √(b^2 - 4ac)) / 2a
Substituting the values of a, b, and c into the quadratic formula, we get:
x = (-(8) ± √((8)^2 - 4(4)(-221))) / 2(4)
x = (-8 ± √(64 + 3536)) / 8
x = (-8 ± √3600) / 8
x = (-8 ± 60) / 8
Now we have the quadratic equation rewritten using the quadratic formula.
Listing the Correct Solutions
To list the correct solutions, we need to simplify the expression and find the values of x.
x = (-8 ± 60) / 8
We can simplify this expression by dividing both terms by 8.
x = -8/8 ± 60/8
x = -1 ± 7.5
Now we have the solutions of the quadratic equation.
x = -1 + 7.5 = 6.5
x = -1 - 7.5 = -8.5
Conclusion
In this article, we have discussed how to solve quadratic equations using the quadratic formula. We have rewritten the given quadratic equation using the quadratic formula and listed the correct solutions. The quadratic formula is a powerful tool for solving quadratic equations, and it can be used to find the solutions of a quadratic equation, even if it cannot be factored easily.
Choosing the Correct Answer
Based on the discussion above, we can choose the correct answer as follows:
- Shows the following quadratic equation correctly rewritten using the quadratic formula:Yes**
- Lists the correct solutions: Yes
Therefore, the correct answer is:
A. x = (-8 ± √(64 + 3536)) / 8
This answer shows the quadratic equation correctly rewritten using the quadratic formula and lists the correct solutions.
Final Answer
Frequently Asked Questions
In this article, we will answer some of the most frequently asked questions about quadratic equations.
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 is two. It has the general form of ax^2 + bx + c = 0, where a, b, and c are constants.
Q: How do I solve a quadratic equation?
A: There are several methods to solve a quadratic equation, including factoring, completing the square, and the quadratic formula. The quadratic formula is a powerful tool for solving quadratic equations and is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
Q: What is the quadratic formula?
A: The quadratic formula is a formula for finding the solutions of a quadratic equation. It is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
Q: How do I use the quadratic formula?
A: To use the quadratic formula, you need to identify the values of a, b, and c in the quadratic equation. Then, you can substitute these values into the quadratic formula and simplify the expression to find the solutions.
Q: What are the steps to solve a quadratic equation using the quadratic formula?
A: The steps to solve a quadratic equation using the quadratic formula are:
- Identify the values of a, b, and c in the quadratic equation.
- Substitute these values into the quadratic formula.
- Simplify the expression to find the solutions.
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. They can be found using the quadratic formula.
Q: Can a quadratic equation have more than two solutions?
A: No, a quadratic equation can have at most two solutions.
Q: Can a quadratic equation have no solutions?
A: Yes, a quadratic equation can have no solutions if the discriminant (b^2 - 4ac) is negative.
Q: What is the discriminant?
A: The discriminant is the expression b^2 - 4ac in the quadratic formula. It determines the nature of the solutions of the quadratic equation.
Q: How do I determine the nature of the solutions of a quadratic equation?
A: To determine the nature of the solutions of a quadratic equation, you need to examine the discriminant. If the discriminant is:
- Positive, the equation has two distinct real solutions.
- Zero, the equation has one real solution.
- Negative, the equation has no real solutions.
Q: Can a quadratic equation have complex solutions?
A: Yes, a quadratic equation can have complex solutions if the discriminant is negative.
Q: How do I find the complex solutions of a quadratic equation?
A: To find the complex solutions of a quadratic equation, you need to use the quadratic formula and simplify the to find the complex values of x.
Conclusion
In this article, we have answered some of the most frequently asked questions about quadratic equations. We have discussed the quadratic formula, the steps to solve a quadratic equation using the quadratic formula, and the nature of the solutions of a quadratic equation. We hope that this article has been helpful in understanding quadratic equations.
Final Answer
The final answer is that quadratic equations are an important topic in mathematics, and understanding them can help you solve a wide range of problems in various fields.