What Are The Solutions To The Equation $x^2 + 6x = 40$?A. $x = -10$ And $x = 4$B. $x = -8$ And $x = 5$C. $x = -5$ And $x = 8$D. $x = -4$ And $x = 10$

Introduction

Solving quadratic equations is a fundamental concept in mathematics, and it is essential to understand the various methods used to find the solutions. In this article, we will focus on solving the equation $x^2 + 6x = 40$ using algebraic methods. We will also explore the different types of solutions that can be obtained, including real and complex solutions.

Understanding the Equation

The given equation is a quadratic equation in the form of $ax^2 + bx + c = 0$, where $a = 1$, $b = 6$, and $c = -40$. To solve this equation, we need to isolate the variable $x$ and find its possible values.

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 $40$ from both sides of the equation:

$x^2 + 6x - 40 = 0$

Factoring the Equation

One of the methods used to solve quadratic equations is factoring. Factoring involves expressing the quadratic expression as a product of two binomials. In this case, we can factor the equation as follows:

$(x + 10)(x - 4) = 0$

Using the Zero Product Property

The zero product property states that if the product of two factors is equal to zero, then at least one of the factors must be equal to zero. We can use this property to find the solutions to the equation:

$x + 10 = 0$ or $x - 4 = 0$

Solving for $x$

We can now solve for $x$ by isolating it in each of the two equations:

$x + 10 = 0 \Rightarrow x = -10$

$x - 4 = 0 \Rightarrow x = 4$

Conclusion

In conclusion, the solutions to the equation $x^2 + 6x = 40$ are $x = -10$ and $x = 4$. These solutions can be obtained by factoring the equation and using the zero product property.

Comparison with Other Options

Let's compare our solutions with the options provided:

A. $x = -10$ and $x = 4$ B. $x = -8$ and $x = 5$ C. $x = -5$ and $x = 8$ D. $x = -4$ and $x = 10$

Our solutions match option A, which is $x = -10$ and $x = 4$.

Real-World Applications

Solving quadratic equations has numerous real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
• Computer Science: Quadratic equations are used in algorithms and data structures to solve problems efficiently.

Complex Solutions

In some cases, quadratic equations may have complex solutions. Complex solutions are solutions that involve imaginary numbers. In this case, the equation $x^2 + 6x = 40$ does not have complex solutions.

Conclusion

In conclusion, solving quadratic equations is a fundamental concept in mathematics that has numerous real-world applications. By understanding the different methods used to solve quadratic equations, we can find the solutions to a wide range of problems.

Final Thoughts

Solving quadratic equations is a skill that requires practice and patience. By mastering this skill, we can solve a wide range of problems and apply mathematical concepts to real-world situations.

References

• [1] "Quadratic Equations" by Math Open Reference
• [2] "Solving Quadratic Equations" by Khan Academy
• [3] "Quadratic Equations in Real-World Applications" by Wolfram MathWorld

Additional Resources

• [1] "Quadratic Equations" by MIT OpenCourseWare
• [2] "Solving Quadratic Equations" by Purplemath
• [3] "Quadratic Equations in Real-World Applications" by Math Is Fun