Solve The Equation:${ -p^2 - 3p - 13 = -2p^2 }$

by ADMIN 49 views

=====================================================

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 a specific quadratic equation, which is given as:

−p2−3p−13=−2p2-p^2 - 3p - 13 = -2p^2

Our goal is to isolate the variable pp and find its possible values. We will use algebraic techniques to simplify the equation and solve for pp.

Understanding the Equation

Before we start solving the equation, let's take a closer look at its structure. The equation is a quadratic equation in the form of ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants. In this case, we have:

−p2−3p−13=−2p2-p^2 - 3p - 13 = -2p^2

We can rewrite the equation as:

−p2−3p−13+2p2=0-p^2 - 3p - 13 + 2p^2 = 0

Simplifying the equation, we get:

p2−3p−13=0p^2 - 3p - 13 = 0

Solving the Equation

To solve the equation, we can use the quadratic formula, which is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In this case, we have a=1a = 1, b=−3b = -3, and c=−13c = -13. Plugging these values into the quadratic formula, we get:

p=−(−3)±(−3)2−4(1)(−13)2(1)p = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-13)}}{2(1)}

Simplifying the expression, we get:

p=3±9+522p = \frac{3 \pm \sqrt{9 + 52}}{2}

p=3±612p = \frac{3 \pm \sqrt{61}}{2}

Finding the Possible Values of pp

Now that we have the quadratic formula, we can find the possible values of pp. We have two possible solutions:

p=3+612p = \frac{3 + \sqrt{61}}{2}

p=3−612p = \frac{3 - \sqrt{61}}{2}

These are the possible values of pp that satisfy the equation.

Conclusion

In this article, we solved a quadratic equation using the quadratic formula. We started with a given equation and simplified it to find the possible values of the variable pp. We used algebraic techniques to isolate the variable and find its possible values. The quadratic formula is a powerful tool for solving quadratic equations, and it is an essential concept in mathematics.

Final Answer

The final answer is:

p=3+612p = \frac{3 + \sqrt{61}}{2}

p=3−612p = \frac{3 - \sqrt{61}}{2}

These are the possible values of pp that satisfy the equation.

Additional Resources

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

Note: The above resources are provided for additional information and are not part of the main content of this article.

=====================================================

Introduction

In our previous article, we solved a quadratic equation using the quadratic formula. However, we know that there are many more questions and doubts that students and professionals may have when it comes to solving quadratic equations. In this article, we will address some of the most frequently asked questions related to solving quadratic equations.

Q&A

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means that the highest power of the variable is two. It is typically written in the form of ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula, you need to identify the values of aa, bb, and cc in the quadratic equation. Then, you plug these values into the quadratic formula and simplify the expression to find the possible values of the variable.

Q: What are the possible values of the variable in a quadratic equation?

A: The possible values of the variable in a quadratic equation are given by the quadratic formula. If the discriminant (b2−4acb^2 - 4ac) is positive, then there are two possible values of the variable. If the discriminant is zero, then there is one possible value of the variable. If the discriminant is negative, then there are no real solutions to the equation.

Q: What is the discriminant in a quadratic equation?

A: The discriminant in a quadratic equation is the expression b2−4acb^2 - 4ac. It is 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 need to calculate the discriminant. If the discriminant is positive, then the equation has two distinct real solutions. If the discriminant is zero, then the equation has one real solution. If the discriminant is negative, then 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 by factoring?

A: Yes, you can solve a quadratic equation by factoring. However, factoring is not always possible, and the quadratic formula is a more general method for solving quadratic equations.

Conclusion

In this article, we addressed some of the most frequently asked questions related to solving quadratic equations. We covered topics such as the quadratic formula, the discriminant, and the nature of the solutions to a quadratic equation. We also discussed the difference between a quadratic equation and a linear equation. We hope that this article has been helpful in clarifying any doubts that you may have had about solving quadratic equations.

Final Answer

The final answer is:

  • The quadratic formula is a mathematical formula that is used to solve quadratic equations.
  • The discriminant is the expression b2−4acb^2 - 4ac in a quadratic equation.
  • The nature of the solutions to a quadratic equation depends on the discriminant.
  • A quadratic equation can be solved by factoring, but the quadratic formula is a more general method.

Additional Resources

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

Note: The above resources are provided for additional information and are not part of the main content of this article.