Complete The Equivalent Equation For − 7 X − 60 = X 2 + 10 X -7x - 60 = X^2 + 10x − 7 X − 60 = X 2 + 10 X . ( X + □ ) ( X + ) = 0 (x + \square)(x + \qquad) = 0 ( X + □ ) ( X + ) = 0 What Are The Solutions Of − 7 X − 60 = X 2 + 10 X -7x - 60 = X^2 + 10x − 7 X − 60 = X 2 + 10 X ? X = □ X = \square X = □ 0

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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 focus on solving the equivalent equation for −7x−60=x2+10x-7x - 60 = x^2 + 10x. We will break down the problem into manageable steps and provide a clear explanation of each step.

Understanding the Problem

The given equation is −7x−60=x2+10x-7x - 60 = x^2 + 10x. Our goal is to rewrite this equation in the form (x+□)(x+)=0(x + \square)(x + \qquad) = 0 and find the solutions for xx.

Step 1: Rearrange the Equation

To start solving the equation, we need to rearrange it to get all the terms on one side. We can do this by adding 7x7x to both sides of the equation and adding 6060 to both sides.

-7x - 60 + 7x + 60 = x^2 + 10x + 7x + 60
0 = x^2 + 17x + 60

Step 2: Factor the Quadratic Expression

Now that we have the equation in the form x2+17x+60=0x^2 + 17x + 60 = 0, we need to factor the quadratic expression. To do this, we need to find two numbers whose product is 6060 and whose sum is 1717. These numbers are 1515 and 44.

x^2 + 17x + 60 = (x + 15)(x + 4) = 0

Step 3: Solve for x

Now that we have the equation in the form (x+15)(x+4)=0(x + 15)(x + 4) = 0, we can solve for xx by setting each factor equal to zero.

x + 15 = 0 \Rightarrow x = -15
x + 4 = 0 \Rightarrow x = -4

Conclusion

In this article, we solved the equivalent equation for −7x−60=x2+10x-7x - 60 = x^2 + 10x by rearranging the equation, factoring the quadratic expression, and solving for xx. We found that the solutions for xx are −15-15 and −4-4.

What are the Solutions of −7x−60=x2+10x-7x - 60 = x^2 + 10x?

The solutions of −7x−60=x2+10x-7x - 60 = x^2 + 10x are x=−15x = \boxed{-15} and x=−4x = \boxed{-4}.

Discussion

Solving quadratic equations is an essential skill in mathematics, and it has numerous applications in various fields such as physics, engineering, and economics. In this article, we provided a step-by-step guide on how to solve the equivalent equation for −7x−60=x2+10x-7x - 60 = x^2 + 10x. We hope that this article has provided valuable insights and knowledge on solving quadratic equations.

Final Thoughts

Solving quadratic equations requires a clear understanding of the concepts and a step-by-step approach. By following the steps outlined in this article, you can solve quadratic equations with ease. Remember to always check your work and verify your solutions to ensure accuracy.

Additional Resources

For more information on solving quadratic equations, we recommend the following resources:

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

Conclusion

In conclusion, solving quadratic equations is a crucial skill in mathematics, and it has numerous applications in various fields. By following the steps outlined in this article, you can solve quadratic equations with ease. Remember to always check your work and verify your solutions to ensure accuracy.

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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 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 ax^2 + bx + c = 0, where a, b, and c are constants.

Q: How Do I Solve a Quadratic Equation?

To solve a quadratic equation, you can use one of the following methods:

  • Factoring: If the quadratic expression can be factored into the product of two binomials, you can set each factor equal to zero and solve for x.
  • Quadratic Formula: If the quadratic expression cannot be factored, you can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.
  • Graphing: You can also graph the quadratic function and find the x-intercepts.

Q: What is the Quadratic Formula?

The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by: x = (-b ± √(b^2 - 4ac)) / 2a.

Q: How Do I Use the Quadratic Formula?

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 Difference Between a Quadratic Equation and a Linear Equation?

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 variable (x^2), while a linear equation does not.

Q: Can I Use the Quadratic Formula to Solve a Linear Equation?

No, you cannot use the quadratic formula to solve a linear equation. The quadratic formula is only used to solve quadratic equations.

Q: What is the Significance of the Discriminant (b^2 - 4ac) in the Quadratic Formula?

The discriminant (b^2 - 4ac) determines the nature of the solutions to a 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?

Yes, you can use the quadratic formula to solve a quadratic equation with complex solutions. However, you need to be careful when simplifying the expression and solving for x.

Q: What is the Relationship Between Quadratic Equations and Graphing?

Quadratic equations can be graphed using a variety of methods, including the use of a graphing calculator or software. The graph of a quadratic equation is a parabola, which is a U-shaped curve.

Conclusion

In this article, we addressed some of the most frequently asked questions about quadratic equations. We hope that this article has provided valuable insights and knowledge on quadratic equations.

Additional Resources

For more information on quadratic equations, we recommend the following resources:

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

Final Thoughts

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By understanding the concepts and techniques outlined in this article, you can solve quadratic equations with ease.