Which Values Of X Are Solutions To The Equation Below? Check All That Apply. 10 X 2 − 56 = 88 − 6 X 2 10x^2 - 56 = 88 - 6x^2 10 X 2 − 56 = 88 − 6 X 2 A. X = − 3 X = -\sqrt{3} X = − 3 B. X = 12 X = 12 X = 12 C. X = 3 X = 3 X = 3 D. X = − 3 X = -3 X = − 3 E. X = 3 X = \sqrt{3} X = 3 F. X = − 12 X = -12 X = − 12
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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 process of solving quadratic equations and apply it to a specific problem. We will also discuss the importance of quadratic equations in real-world applications.
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, the quadratic formula, and graphing.
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 a quadratic equation, even if it cannot be factored easily.
Solving the Given Equation
Now, let's apply the quadratic formula to the given equation:
10x^2 - 56 = 88 - 6x^2
First, we need to simplify the equation by combining like terms:
10x^2 + 6x^2 = 88 + 56
This gives us:
16x^2 = 144
Next, we can divide both sides of the equation by 16 to isolate x^2:
x^2 = 144/16
x^2 = 9
Now, we can take the square root of both sides of the equation to find the solutions:
x = ±√9
x = ±3
Checking the Solutions
Now that we have found the solutions, we need to check if they are correct. We can do this by plugging the solutions back into the original equation:
10x^2 - 56 = 88 - 6x^2
For x = 3:
10(3)^2 - 56 = 88 - 6(3)^2
90 - 56 = 88 - 54
34 = 34
This is true, so x = 3 is a solution.
For x = -3:
10(-3)^2 - 56 = 88 - 6(-3)^2
90 - 56 = 88 - 54
34 = 34
This is also true, so x = -3 is a solution.
Conclusion
In this article, we have solved a quadratic equation using the quadratic formula and checked the solutions. We have also discussed the importance of quadratic equations in real-world applications. Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike.
Final Answer
Based on the solutions we have found, the correct answers are:
- A. is not a solution
- B. is not a solution
- C. is a solution
- D. is a solution
- E. is not a solution
- F. is not a solution
Note: The original problem statement asks to check all that apply, but based on our solution, only C and D are correct.
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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 our previous article, we explored the process of solving quadratic equations and applied it to a specific problem. In this article, we will answer some frequently asked questions about quadratic equations.
Q&A
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: There are several methods to solve a quadratic equation, including:
- Factoring: If the quadratic equation can be factored easily, 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 it on a coordinate plane.
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 plug in the values of a, b, and c into the formula. Then, you can simplify the expression and find the solutions.
Q: What are the solutions to a quadratic equation?
A: The solutions to a quadratic equation are the values of x that make the equation true. In other words, they are the values of x that satisfy the equation.
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 always gives two solutions, and there is no way to have more than two solutions.
Q: Can a quadratic equation have no solutions?
A: Yes, a quadratic equation can have no solutions. This happens when 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 to the quadratic equation.
Q: How do I determine the nature of the solutions to a quadratic equation?
A: To determine the nature of the solutions to a quadratic equation, you need to check 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.
Conclusion
In this article, we have answered some frequently asked questions about quadratic equations. We have discussed the definition of a quadratic equation, the methods to solve it, and the nature of the solutions. Quadratic equations are a fundamental concept in mathematics, and understanding them is crucial for students and professionals alike.
Final Tips
- Always check the discriminant to determine the nature of the solutions to a quadratic equation.
- Use the quadratic formula to find the solutions to a quadratic equation.
- Graphing can also be used to solve a quadratic equation, but it is not as efficient as the quadratic formula.
Note: The above Q&A is based on the general form of a quadratic equation, ax^2 + bx + c = 0. If you have a specific quadratic equation, you may need to adjust the answers accordingly.