{ \left(x+\frac{13}{2}\right)\left(x-\frac{13}{2}\right)=0$}$How Many Distinct Real Solutions Does The Given Equation Have?Choose One Answer:(A) 0 (B) 1 (C) 2 (D) 3

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 focus on solving quadratic equations of the form $\left(x+\frac{13}{2}\right)\left(x-\frac{13}{2}\right)=0$. We will explore the concept of quadratic equations, the different methods of solving them, and the number of distinct real solutions they have.

What are Quadratic Equations?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $x$) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a$ cannot be equal to zero. Quadratic equations can be solved using various methods, including factoring, the quadratic formula, and graphing.

The Given Equation

The given equation is $\left(x+\frac{13}{2}\right)\left(x-\frac{13}{2}\right)=0$. This equation can be solved by factoring, which involves expressing the equation as a product of two binomials. In this case, the equation can be factored as:

$$\left(x+\frac{13}{2}\right)\left(x-\frac{13}{2}\right)=0$$

Solving the Equation

To solve the equation, we need to find the values of $x$ that make the equation true. We can do this by setting each factor equal to zero and solving for $x$. In this case, we have:

$$x+\frac{13}{2}=0 \quad \text{or} \quad x-\frac{13}{2}=0$$

Solving for $x$ in each equation, we get:

$$x=-\frac{13}{2} \quad \text{or} \quad x=\frac{13}{2}$$

Number of Distinct Real Solutions

Now that we have found the solutions to the equation, we need to determine the number of distinct real solutions. A distinct real solution is a value of $x$ that makes the equation true and is not equal to any other solution. In this case, we have two solutions: $x=-\frac{13}{2}$ and $x=\frac{13}{2}$. Since these two solutions are distinct and real, the given equation has two distinct real solutions.

Conclusion

In conclusion, the given equation $\left(x+\frac{13}{2}\right)\left(x-\frac{13}{2}\right)=0$ has two distinct real solutions: $x=-\frac{13}{2}$ and $x=\frac{13}{2}$. We can solve quadratic equations using various methods, including factoring, the quadratic formula, and graphing. By understanding the concept of quadratic equations and the different methods of solving them, we can solve a wide range of problems in mathematics and other fields.

Final Answer

The final answer is $\boxed{2}$.

Additional Resources

For more information on quadratic equations and how to solve them, check out the following resources:

• Khan Academy: Quadratic Equations
• Mathway: Quadratic Equation Solver
• Wolfram Alpha: Quadratic Equation Solver

Frequently Asked Questions

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 (in this case, $x$) is two.

Q: How do I solve a quadratic equation? A: You can solve a quadratic equation using various methods, including factoring, the quadratic formula, and graphing.

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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 explored the concept of quadratic equations, the different methods of solving them, and the number of distinct real solutions they have. In this article, we will provide a comprehensive Q&A guide on quadratic equations, covering various topics and concepts.

Q&A Section

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 (in this case, $x$) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a$ cannot be equal to zero.

Q: How do I solve a quadratic equation?

A: You can solve a quadratic equation using various methods, including factoring, the quadratic formula, and graphing. Factoring involves expressing the equation as a product of two binomials, while the quadratic formula involves using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the solutions.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula used to find the solutions of a quadratic equation. 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. Then, simplify the expression and solve for $x$.

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: What does the discriminant tell us?

A: The discriminant tells us whether the quadratic equation has real or complex solutions. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has two complex solutions.

Q: How do I determine the number of distinct real solutions?

A: To determine the number of distinct real solutions, you need to examine the discriminant. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one distinct real solution. If the discriminant is negative, the equation has no real solutions.

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. A quadratic equation has a parabolic shape, while a linear equation has a straight line shape.

Q: Can I solve a quadratic equation using graphing?

A: Yes, you can solve a quadratic equation using graphing. By graphing the quadratic equation, you can find the x-intercepts, which represent the solutions of the equation.

Q: What are some common applications of quadratic equations?

A: Quadratic equations have many applications in various fields, including physics, engineering, and economics. Some common applications include projectile motion, optimization problems, and quadratic programming.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and they have many applications in various fields. By understanding the concept of quadratic equations and the different methods of solving them, you can solve a wide range of problems in mathematics and other fields. We hope this Q&A guide has provided you with a comprehensive understanding of quadratic equations.

Additional Resources

For more information on quadratic equations and how to solve them, check out the following resources:

• Khan Academy: Quadratic Equations
• Mathway: Quadratic Equation Solver
• Wolfram Alpha: Quadratic Equation Solver

Frequently Asked Questions

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 (in this case, $x$) is two.

Q: How do I solve a quadratic equation? A: You can solve a quadratic equation using various methods, including factoring, the quadratic formula, and graphing.

Q: What is the quadratic formula? A: The quadratic formula is a mathematical formula used to find the solutions of a 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, simplify the expression and solve for $x$.

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: What does the discriminant tell us? A: The discriminant tells us whether the quadratic equation has real or complex solutions. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has two complex solutions.