What Are The Solutions Of $x^2 + 4x = 9$?

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Introduction

Solving quadratic equations is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, and economics. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. In this article, we will focus on solving the quadratic equation $x^2 + 4x = 9$.

Understanding the Equation

The given equation is a quadratic equation in the form of $ax^2 + bx + c = 0$, where $a = 1$, $b = 4$, and $c = -9$. To solve this equation, we need to isolate the variable $x$.

Rearranging the Equation

The first step in solving the equation is to rearrange it in the standard form of a quadratic equation. We can do this by subtracting 9 from both sides of the equation:

x2+4x−9=0x^2 + 4x - 9 = 0

Factoring the Equation

One way to solve a quadratic equation is to factor it. However, in this case, the equation does not factor easily. Therefore, we will use the quadratic formula to solve the equation.

The Quadratic Formula

The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by:

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

In this case, $a = 1$, $b = 4$, and $c = -9$. Plugging these values into the formula, we get:

x=−4±42−4(1)(−9)2(1)x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-9)}}{2(1)}

Simplifying the Equation

Simplifying the equation, we get:

x=−4±16+362x = \frac{-4 \pm \sqrt{16 + 36}}{2}

x=−4±522x = \frac{-4 \pm \sqrt{52}}{2}

x=−4±2132x = \frac{-4 \pm 2\sqrt{13}}{2}

x=−2±13x = -2 \pm \sqrt{13}

Solutions of the Equation

Therefore, the solutions of the equation $x^2 + 4x = 9$ are:

x=−2+13x = -2 + \sqrt{13}

x=−2−13x = -2 - \sqrt{13}

Conclusion

In this article, we have solved the quadratic equation $x^2 + 4x = 9$ using the quadratic formula. The solutions of the equation are $x = -2 + \sqrt{13}$ and $x = -2 - \sqrt{13}$. These solutions can be verified by plugging them back into the original equation.

Applications of Quadratic Equations

Quadratic equations have numerous applications in various fields such as physics, engineering, and economics. Some of the applications of quadratic equations include:

  • Projectile Motion: Quadratic equations are used to model the trajectory of a projectile under the influence of gravity.
  • Optimization: Quadratic equations are used to optimize functions and find the maximum or minimum value of a function.
  • Electrical Circuits: Quadratic equations are used to analyze and design electrical circuits.
  • Computer Graphics: Quadratic equations are used to create 3D models and animations.

Tips for Solving Quadratic Equations

Here are some tips for solving quadratic equations:

  • Use the Quadratic Formula: The quadratic formula is a powerful tool for solving quadratic equations.
  • Factor the Equation: If the equation can be factored easily, it is often faster to factor it than to use the quadratic formula.
  • Check Your Solutions: Always check your solutions by plugging them back into the original equation.
  • Use a Calculator: If you are having trouble solving a quadratic equation, try using a calculator to check your solutions.

Common Quadratic Equations

Here are some common quadratic equations:

  • x^2 + 4x + 4 = 0: This equation can be factored as (x + 2)^2 = 0.
  • x^2 - 4x + 4 = 0: This equation can be factored as (x - 2)^2 = 0.
  • x^2 + 2x - 3 = 0: This equation can be factored as (x + 3)(x - 1) = 0.

Conclusion

In conclusion, solving quadratic equations is a fundamental concept in mathematics. The quadratic formula is a powerful tool for solving quadratic equations, and it has numerous applications in various fields. By following the tips and examples provided in this article, you can become proficient in solving quadratic equations and apply them to real-world problems.

Introduction

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields such as physics, engineering, and economics. In our previous article, we solved the quadratic equation $x^2 + 4x = 9$ using the quadratic formula. In this article, we will answer some 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. It is in the form of $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic equation?

A: There are several ways to solve a quadratic equation, including:

  • Factoring: If the equation can be factored easily, it is often faster to factor it than to use the quadratic formula.
  • Quadratic Formula: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
  • Graphing: You can also solve a quadratic equation by graphing the related function and finding the x-intercepts.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

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 are the solutions of a quadratic equation?

A: The solutions of a quadratic equation are the values of $x$ that make the equation true. They can be real or complex numbers.

Q: How do I check my solutions?

A: To check your solutions, plug them back into the original equation and simplify. If the equation is true, then the solution is correct.

Q: What are some common quadratic equations?

A: Some common quadratic equations include:

  • x^2 + 4x + 4 = 0: This equation can be factored as (x + 2)^2 = 0.
  • x^2 - 4x + 4 = 0: This equation can be factored as (x - 2)^2 = 0.
  • x^2 + 2x - 3 = 0: This equation can be factored as (x + 3)(x - 1) = 0.

Q: What are some real-world applications of quadratic equations?

A: Quadratic equations have numerous applications in various fields such as physics, engineering, and economics. Some of the applications include:

  • Projectile Motion: Quadratic equations are used to model the trajectory of a projectile under the influence of gravity.
  • Optimization: Quadratic equations are used to optimize functions and find the maximum or minimum value of a function.
  • Electrical Circuits: Quadratic equations are used to analyze and design electrical circuits.
  • Computer Graphics: Quadratic equations are used to create 3D models and animations.

Q: What are some tips for solving quadratic equations?

A: Here are some tips for solving quadratic equations:

  • Use the Quadratic Formula: The quadratic formula is a powerful tool for solving quadratic equations.
  • Factor the Equation: If the equation can be factored easily, it is often faster to factor it than to use the quadratic formula.
  • Check Your Solutions: Always check your solutions by plugging them back into the original equation.
  • Use a Calculator: If you are having trouble solving a quadratic equation, try using a calculator to check your solutions.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields. By understanding the quadratic formula and how to use it, you can solve quadratic equations and apply them to real-world problems. We hope this Q&A article has been helpful in answering your questions about quadratic equations.