Which Graph Correctly Solves The Equation Below?\[$-x^2 - 1 = 2x^2 - 4\$\]A. B.

Mar 1, 2025 by ADMIN 81 views

**Solving Quadratic Equations: A Graphical Approach**

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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 how to solve a quadratic equation using a graphical approach. We will examine the given equation, $-x^2 - 1 = 2x^2 - 4$ , and determine which graph correctly solves it.

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. Quadratic equations can be solved using various methods, including factoring, the quadratic formula, and graphical methods.

The Given Equation

The given equation is $-x^2 - 1 = 2x^2 - 4$ . To solve this equation, we need to isolate the variable $x$ . We can start by adding $1$ to both sides of the equation, which gives us $-x^2 = 2x^2 - 5$ .

Rearranging the Equation

Next, we can add $x^2$ to both sides of the equation, which gives us $0 = 3x^2 - 5$ . Now, we can add $5$ to both sides of the equation, which gives us $5 = 3x^2$ .

Solving for x

To solve for $x$ , we can divide both sides of the equation by $3$ , which gives us $\frac{5}{3} = x^2$ . Now, we can take the square root of both sides of the equation, which gives us $x = \pm \sqrt{\frac{5}{3}}$ .

Graphical Approach

To visualize the solution, we can graph the equation $-x^2 - 1 = 2x^2 - 4$ . We can start by plotting the two functions $y = -x^2 - 1$ and $y = 2x^2 - 4$ on the same coordinate plane.

Graph A

Graph A represents the equation $y = -x^2 - 1$ . This is a downward-facing parabola with a vertex at $(0, -1)$ .

Graph B

Graph B represents the equation $y = 2x^2 - 4$ . This is an upward-facing parabola with a vertex at $(0, -4)$ .

Comparing the Graphs

To determine which graph correctly solves the equation, we need to compare the two graphs. We can see that Graph A is a downward-facing parabola, while Graph B is an upward-facing parabola. Since the given equation is $-x^2 - 1 = 2x^2 - 4$ , we know that the solution must be a downward-facing parabola.

Conclusion

Based on our analysis, we can conclude that Graph A is the correct solution to the equation $-x^2 - 1 = 2x^2 - 4$ . Graph A represents a downward-facing parabola, which is consistent with the given equation. Graph B, on the other hand, represents an upward-facing parabola, which is not consistent with the given equation.

Final Answer

The final answer is Graph A.

Additional Resources

For more information on solving quadratic equations, please refer to the following resources:

References

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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 address 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. 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: There are several methods to solve a quadratic equation, including factoring, the quadratic formula, and graphical methods. The method you choose will depend on the specific equation and your personal preference.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that can be used to solve quadratic equations. The formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , 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. You will then get two solutions for $x$ , which are the values that satisfy the equation.

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 I solve a quadratic equation by graphing?

A: Yes, you can solve a quadratic equation by graphing. By plotting the two functions $y = ax^2 + bx + c$ and $y = 0$ on the same coordinate plane, you can find the points where the two functions intersect. These points will be the solutions to the equation.

Q: What is the vertex of a quadratic equation?

A: The vertex of a quadratic equation is the point on the graph where the function changes from decreasing to increasing or vice versa. The vertex is also the minimum or maximum point of the function.

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

A: To find the vertex of a quadratic equation, you can use the formula $x = -\frac{b}{2a}$ . This will give you the x-coordinate of the vertex. You can then plug this value into the equation to find the y-coordinate of the vertex.

Q: Can I solve a quadratic equation with complex numbers?

A: Yes, you can solve a quadratic equation with complex numbers. In fact, complex numbers are often used to solve quadratic equations that have no real solutions.

Q: What is the discriminant of a quadratic equation?

A: The discriminant of a quadratic equation is the expression $b^2 - 4ac$ that appears in the quadratic formula. The discriminant can be used to determine the nature of the solutions to the 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 can use the discriminant. 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.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we have addressed some of the most frequently asked questions about quadratic equations. We hope that this information has been helpful and informative.

Final Answer

The final answer is that quadratic equations are a powerful tool for solving mathematical problems, and understanding them is essential for success in mathematics and other fields.

Additional Resources

For more information on quadratic equations, please refer to the following resources: