Select ALL The Correct Answers.Which Of The Following Statements Are True About The Equation Below?$x^2 - 6x + 2 = 0$A. The Extreme Value Is At The Point $(3, -7$\].B. The Extreme Value Is At The Point $(7, -3$\].C. The

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 this article, we will explore the properties of quadratic equations, specifically the equation $x^2 - 6x + 2 = 0$. We will examine the characteristics of this equation, including its extreme values, and determine which of the given statements are true.

Understanding Quadratic Equations

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. In the given equation, $x^2 - 6x + 2 = 0$, the coefficients are $a = 1$, $b = -6$, and $c = 2$.

Finding the Extreme Values

The extreme values of a quadratic equation occur at the vertex of the parabola. To find the vertex, we can use the formula $x = -\frac{b}{2a}$. In this case, $a = 1$ and $b = -6$, so the x-coordinate of the vertex is $x = -\frac{-6}{2(1)} = 3$.

To find the y-coordinate of the vertex, we can substitute the x-coordinate into the equation: $y = (3)^2 - 6(3) + 2 = 9 - 18 + 2 = -7$. Therefore, the vertex of the parabola is at the point $(3, -7)$.

Evaluating the Statements

Now that we have found the vertex of the parabola, we can evaluate the given statements.

• A. The extreme value is at the point $(3, -7)$. This statement is true, as we have just shown that the vertex of the parabola is at the point $(3, -7)$.
• B. The extreme value is at the point $(7, -3)$. This statement is false, as we have found that the vertex of the parabola is at the point $(3, -7)$, not $(7, -3)$.
• C. The equation has two real roots. This statement is false, as the equation $x^2 - 6x + 2 = 0$ has only one real root, which is the x-coordinate of the vertex, $x = 3$.

Conclusion

In conclusion, the equation $x^2 - 6x + 2 = 0$ has a vertex at the point $(3, -7)$. The extreme value is at this point, and the equation has only one real root, which is the x-coordinate of the vertex. Therefore, statement A is true, and statements B and C are false.

Step 1: Write the Quadratic Equation

The first step in solving a quadratic equation is to write it in the general form $ax^2 + bx + c = 0$. In this case, the equation is $x^2 - 6x + 2 = 0$.

Step 2: Find the Vertex

To find the vertex of the parabola, we can use the formula $x = -\frac{b}{2a}$. In this case, $a = 1$ and $b = -6$, so the x-coordinate of the vertex is $x = -\frac{-6}{2(1)} = 3$.

Step 3: Find the y-Coordinate of the Vertex

To find the y-coordinate of the vertex, we can substitute the x-coordinate into the equation: $y = (3)^2 - 6(3) + 2 = 9 - 18 + 2 = -7$. Therefore, the vertex of the parabola is at the point $(3, -7)$.

Step 4: Evaluate the Statements

Now that we have found the vertex of the parabola, we can evaluate the given statements.

• A. The extreme value is at the point $(3, -7)$. This statement is true, as we have just shown that the vertex of the parabola is at the point $(3, -7)$.
• B. The extreme value is at the point $(7, -3)$. This statement is false, as we have found that the vertex of the parabola is at the point $(3, -7)$, not $(7, -3)$.
• C. The equation has two real roots. This statement is false, as the equation $x^2 - 6x + 2 = 0$ has only one real root, which is the x-coordinate of the vertex, $x = 3$.

Step 5: Conclusion

In conclusion, the equation $x^2 - 6x + 2 = 0$ has a vertex at the point $(3, -7)$. The extreme value is at this point, and the equation has only one real root, which is the x-coordinate of the vertex. Therefore, statement A is true, and statements B and C are false.

Frequently Asked Questions

Q: What is the vertex of the parabola?

A: The vertex of the parabola is at the point $(3, -7)$.

Q: How many real roots does the equation have?

A: The equation has only one real root, which is the x-coordinate of the vertex, $x = 3$.

Q: Which of the given statements are true?

A: Statement A is true, and statements B and C are false.

Conclusion

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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 this article, we will provide a comprehensive Q&A guide to quadratic equations, covering topics such as solving quadratic equations, finding the vertex, and evaluating statements.

Q&A Guide

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. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Alternatively, you can factor the equation or complete the square to find the solutions.

Q: What is the vertex of a parabola?

A: The vertex of a parabola is the point at which the parabola changes direction. It is the minimum or maximum point of the parabola, depending on the direction of the parabola.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you can use the formula $x = -\frac{b}{2a}$. This will give you the x-coordinate of the vertex. To find the y-coordinate, you can substitute the x-coordinate into the equation.

Q: What is the significance of the vertex?

A: The vertex is significant because it represents the extreme value of the parabola. It is the point at which the parabola changes direction, and it is the minimum or maximum point of the parabola.

Q: How do I evaluate statements about a quadratic equation?

A: To evaluate statements about a quadratic equation, you need to analyze the equation and determine whether the statement is true or false. You can use the properties of the equation, such as the vertex and the solutions, to evaluate the statement.

Q: What are some common mistakes to avoid when working with quadratic equations?

A: Some common mistakes to avoid when working with quadratic equations include:

• Not following the order of operations
• Not simplifying the equation
• Not using the correct formula for the solutions
• Not checking the solutions for validity

Q: How do I apply quadratic equations in real-world situations?

A: Quadratic equations have many applications in real-world situations, such as:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
• Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and they have many applications in real-world situations. By understanding the properties of quadratic equations, such as the vertex and the solutions, you can evaluate statements and apply quadratic equations in a variety of contexts.

Frequently Asked Questions

Q: What is the quadratic formula?

A: The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: How do I factor a quadratic equation?

A: To factor a quadratic equation, you need to find two numbers whose product is $ac$ and whose sum is $b$. These numbers are the factors of the equation.

Q: What is the significance of the discriminant?

A: The discriminant is the expression $b^2 - 4ac$ under the square root in the quadratic formula. It determines the nature of the solutions to the equation.

Q: How do I complete the square?

A: To complete the square, you need to add and subtract the square of half the coefficient of the $x$ term to the equation.

Common Quadratic Equations

Q: What is the equation of a parabola with vertex at $(h, k)$?

A: The equation of a parabola with vertex at $(h, k)$ is $y = a(x - h)^2 + k$.

Q: What is the equation of a parabola with axis of symmetry $x = h$?

A: The equation of a parabola with axis of symmetry $x = h$ is $y = a(x - h)^2 + k$.

Q: What is the equation of a parabola with vertex at $(h, k)$ and axis of symmetry $x = h$?

A: The equation of a parabola with vertex at $(h, k)$ and axis of symmetry $x = h$ is $y = a(x - h)^2 + k$.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and they have many applications in real-world situations. By understanding the properties of quadratic equations, such as the vertex and the solutions, you can evaluate statements and apply quadratic equations in a variety of contexts.