Complete The Equivalent Equation For \[$-7x - 60 = X^2 + 10x\$\].\[$(x + \checkmark)(x + \checkmark \vee) = 0\$\]What Are The Solutions Of \[$-7x - 60 = X^2 + 10x\$\]?\[$x = \square\$\]

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 focus on solving the equivalent equation ${-7x - 60 = x^2 + 10x}$. We will break down the solution process into manageable steps and provide a clear explanation of each step.

Understanding the Equation

The given equation is a quadratic equation in the form of ${ax^2 + bx + c = 0}$. To solve this equation, we need to first rewrite it in the standard form by moving all terms to one side of the equation.

${-7x - 60 = x^2 + 10x}$

To rewrite the equation, we can add ${7x}$ to both sides and add ${60}$ to both sides.

${x^2 + 10x + 7x + 60 = 0}$

Combining like terms, we get:

${x^2 + 17x + 60 = 0}$

Factoring the Quadratic Equation

Now that we have the equation in the standard form, we can try to factor it. Factoring a quadratic equation involves expressing it as a product of two binomials.

To factor the equation ${x^2 + 17x + 60 = 0}$, we need to find two numbers whose product is ${60}$ and whose sum is ${17}$. These numbers are ${15}$ and ${4}$, since ${15 \times 4 = 60}$ and ${15 + 4 = 19}$.

However, we need to find numbers that add up to ${17}$, not ${19}$. Let's try again. The numbers ${12}$ and ${5}$ have a product of ${60}$ and a sum of ${17}$.

So, we can write the equation as:

${(x + 12)(x + 5) = 0}$

Solving for x

Now that we have factored the equation, we can solve for ${x}$ by setting each factor equal to zero.

${x + 12 = 0}$

Subtracting ${12}$ from both sides, we get:

${x = -12}$

${x + 5 = 0}$

Subtracting ${5}$ from both sides, we get:

${x = -5}$

Conclusion

In this article, we solved the equivalent equation ${-7x - 60 = x^2 + 10x}$ by rewriting it in the standard form, factoring it, and solving for ${x}$. We found that the solutions to the equation are ${x = -12}$ and ${x = -5}$.

Final Answer

The final answer is:

${x = \boxed{-12, -5}}$

Discussion

This problem is a great example of how to solve quadratic equations by factoring. The key is to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term. With practice, you will become more comfortable with factoring quadratic equations and solving for ${x}$.

Related Problems

If you want to practice solving quadratic equations, try the following problems:

• ${x^2 + 14x + 48 = 0}$
• ${x^2 - 7x - 18 = 0}$
• ${x^2 + 2x - 15 = 0}$

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task for many students and professionals. In this article, we will address some of the most frequently asked questions about quadratic equations and provide clear explanations and examples to help you understand the concepts better.

Q: What is a quadratic equation?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two. It is typically 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?

There are several methods to solve a quadratic equation, including factoring, using the quadratic formula, and completing the square. The method you choose depends on the form of the equation and your personal preference.

Q: What is the quadratic formula?

The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is written as:

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

This formula can be used to solve any quadratic equation, regardless of whether it can be factored or not.

Q: What is the difference between the quadratic formula and factoring?

The quadratic formula and factoring are two different methods of solving quadratic equations. Factoring involves expressing the equation as a product of two binomials, while the quadratic formula involves using a mathematical formula to find the solutions.

Q: When should I use the quadratic formula?

You should use the quadratic formula when the equation cannot be factored easily or when you are dealing with a complex equation. The quadratic formula is a powerful tool that can be used to solve any quadratic equation, regardless of its complexity.

Q: What is the discriminant?

The discriminant is a value that is used in the quadratic formula to determine the nature of the solutions. It is calculated as ${b^2 - 4ac}$. 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: How do I determine the nature of the solutions?

To determine the nature of the solutions, you need to calculate the discriminant and examine its value. 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: What are the applications of quadratic equations?

Quadratic equations have numerous applications in various fields, including physics, engineering, economics, and computer science. They are used to model real-world problems, such as the motion of objects, the growth of populations, and the behavior of electrical circuits.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the concepts and techniques involved. In this article, we have addressed some of the most frequently asked questions about quadratic equations and provided clear explanations and examples to help you understand the concepts better.

Final Tips

• Practice, practice, practice: The more you practice solving quadratic equations, the more comfortable you will become with the concepts and techniques involved.
• Use the quadratic formula when you are dealing with complex equations or when factoring is not possible.
• Calculate the discriminant to determine the nature of the solutions.
• Apply quadratic equations to real-world problems to see their practical applications.

Related Resources

If you want to learn more about quadratic equations, check out the following resources:

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

Remember, solving quadratic equations is a skill that takes time and practice to develop. With persistence and dedication, you can become proficient in solving quadratic equations and apply them to real-world problems.