Which Of The Following Reveals The Minimum Value For The Equation 2 X 2 − 4 X − 2 = 0 2x^2 - 4x - 2 = 0 2 X 2 − 4 X − 2 = 0 ?A. 2 ( X − 1 ) 2 = 4 2(x-1)^2 = 4 2 ( X − 1 ) 2 = 4 B. 2 ( X − 1 ) 2 = − 4 2(x-1)^2 = -4 2 ( X − 1 ) 2 = − 4 C. 2 ( X − 2 ) 2 = 4 2(x-2)^2 = 4 2 ( X − 2 ) 2 = 4 D. 2 ( X − 2 ) 2 = − 4 2(x-2)^2 = -4 2 ( X − 2 ) 2 = − 4

Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. In this article, we will focus on solving quadratic equations and finding the minimum value for a given equation.

Understanding Quadratic Equations

A quadratic equation is generally written in the form of $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. The minimum value of a quadratic equation can be found using various methods, including factoring, completing the square, and using the quadratic formula.

Factoring Quadratic Equations

Factoring quadratic equations involves expressing the equation as a product of two binomials. This method is useful when the equation can be easily factored. However, not all quadratic equations can be factored, and in such cases, other methods must be used.

Completing the Square

Completing the square is a method used to solve quadratic equations by rewriting the equation in a perfect square trinomial form. This method involves adding and subtracting a constant term to create a perfect square trinomial.

Quadratic Formula

The quadratic formula is a method used to solve quadratic equations when the equation cannot be factored or completed to a perfect square. The quadratic formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Solving the Given Equation

The given equation is $2x^2 - 4x - 2 = 0$. To find the minimum value of this equation, we can use the method of completing the square.

Step 1: Divide the equation by 2

Dividing the equation by 2 gives us:

$x^2 - 2x - 1 = 0$

Step 2: Add 1 to both sides of the equation

Adding 1 to both sides of the equation gives us:

$x^2 - 2x = 1$

Step 3: Add $(\frac{-b}{2})^2$ to both sides of the equation

Adding $(\frac{-b}{2})^2$ to both sides of the equation gives us:

$x^2 - 2x + 1 = 1 + 1$

Step 4: Factor the left-hand side of the equation

Factoring the left-hand side of the equation gives us:

$(x - 1)^2 = 2$

Step 5: Multiply both sides of the equation by 2

Multiplying both sides of the equation by 2 gives us:

$2(x - 1)^2 = 4$

Conclusion

In conclusion, the minimum value for the equation $2x^2 - 4x - 2 = 0$ is revealed by option A: $2(x-1)^2 = 4$. This is because the equation can be rewritten as $(x - 1)^2 = 2$, and multiplying both sides by 2 gives us $2(x - 1)^2 = 4$.

Discussion

The given equation can be solved using various methods, including factoring, completing the square, and using the quadratic formula. However, the method of completing the square is the most straightforward and efficient method in this case.

Comparison of Options

Let's compare the given options:

• Option A: $2(x-1)^2 = 4$
• Option B: $2(x-1)^2 = -4$
• Option C: $2(x-2)^2 = 4$
• Option D: $2(x-2)^2 = -4$

Option A is the correct answer because it is the result of completing the square method. Options B, C, and D are incorrect because they do not represent the minimum value of the equation.

Final Answer

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Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In our previous article, we discussed how to solve quadratic equations and find the minimum value for a given equation. In this article, we will provide a Q&A guide to help you better understand 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 is generally written in the form of $ax^2 + bx + c = 0$, 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 quadratic equations, including factoring, completing the square, and using the quadratic formula. The method you choose depends on the equation and your personal preference.

Q: What is the quadratic formula?

A: The quadratic formula is a method used to solve quadratic equations when the equation cannot be factored or completed to a perfect square. The quadratic formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Q: How do I find the minimum value of a quadratic equation?

A: To find the minimum value of a quadratic equation, you can use the method of completing the square. This involves rewriting the equation in a perfect square trinomial form.

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 squared term, while a linear equation does not.

Q: Can I use the quadratic formula to solve a quadratic equation that can be factored?

A: Yes, you can use the quadratic formula to solve a quadratic equation that can be factored. However, it is generally more efficient to factor the equation instead.

Q: How do I determine if a quadratic equation can be factored?

A: To determine if a quadratic equation can be factored, you can try to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.

Q: What is the significance of the discriminant in the quadratic formula?

A: The discriminant is the expression under the square root in the quadratic formula, which is given by $b^2 - 4ac$. The discriminant determines the nature of the solutions to the quadratic equation. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

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 two complex solutions, which can be written in the form of $x = a \pm bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. By understanding how to solve quadratic equations and find the minimum value for a given equation, you can better appreciate the beauty and power of mathematics. We hope this Q&A guide has been helpful in answering your questions and providing a deeper understanding of quadratic equations.

Frequently Asked Questions

• What is a quadratic equation?
• How do I solve a quadratic equation?
• What is the quadratic formula?
• How do I find the minimum value of a quadratic equation?
• What is the difference between a quadratic equation and a linear equation?
• Can I use the quadratic formula to solve a quadratic equation that can be factored?
• How do I determine if a quadratic equation can be factored?
• What is the significance of the discriminant in the quadratic formula?
• Can I use the quadratic formula to solve a quadratic equation with complex solutions?

Answers

• A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two.
• There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula.
• The quadratic formula is a method used to solve quadratic equations when the equation cannot be factored or completed to a perfect square.
• To find the minimum value of a quadratic equation, you can use the method of completing the square.
• A quadratic equation has a squared term, while a linear equation does not.
• Yes, you can use the quadratic formula to solve a quadratic equation that can be factored.
• To determine if a quadratic equation can be factored, you can try to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.
• The discriminant determines the nature of the solutions to the quadratic equation.
• Yes, you can use the quadratic formula to solve a quadratic equation with complex solutions.