Solving $x^2 + 2x = -10$ A Step-by-Step Guide

In this comprehensive article, we will delve into the solution of the quadratic equation $x^2 + 2x = -10$. This problem falls under the domain of algebra, a fundamental branch of mathematics. Quadratic equations, characterized by the highest power of the variable being 2, are prevalent in various mathematical and real-world applications. Solving them accurately is a crucial skill. We will explore different methods to find the solution, emphasizing the correct approach and explaining why certain options are valid while others are not. We aim to provide a clear, step-by-step explanation that will help readers understand not just the solution to this particular equation, but also the general techniques for solving quadratic equations.

Understanding Quadratic Equations

Before diving into the specific problem, it's essential to understand what quadratic equations are and the standard forms they take. A quadratic equation is generally represented as $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable we aim to solve for. The key characteristic is the presence of the $x^2$ term, which distinguishes it from linear equations. Quadratic equations can have two solutions, one solution, or no real solutions, depending on the discriminant ($b^2 - 4ac$). These solutions are also known as roots or zeros of the quadratic equation. Understanding the nature of roots is crucial in various mathematical contexts, including graphing parabolas and solving optimization problems. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Each method has its strengths and is suitable for different types of quadratic equations. For instance, factoring is efficient when the equation can be easily factored, while the quadratic formula is a universal method applicable to all quadratic equations.

Transforming the Equation

The first step in solving the given equation, $x^2 + 2x = -10$, is to rewrite it in the standard quadratic form, which is $ax^2 + bx + c = 0$. This involves moving all terms to one side of the equation, leaving zero on the other side. To do this, we add 10 to both sides of the equation:

$x^2 + 2x + 10 = 0$

Now, the equation is in the standard form, where $a = 1$, $b = 2$, and $c = 10$. This transformation is crucial because it allows us to apply standard methods for solving quadratic equations, such as the quadratic formula or completing the square. Recognizing the standard form is the foundation for any further steps in solving quadratic equations. It helps in identifying the coefficients, which are essential for applying different solution techniques. This initial step of transforming the equation is often overlooked, but it is a critical step in simplifying the problem and making it solvable using known methods. Furthermore, expressing the equation in standard form makes it easier to analyze the nature of the roots using the discriminant, which we will discuss later.

Applying the Quadratic Formula

Since the equation $x^2 + 2x + 10 = 0$ does not factor easily, the most reliable method to find the solutions is the quadratic formula. The quadratic formula is a general formula that provides the solutions to any quadratic equation in the form $ax^2 + bx + c = 0$. The formula is given by:

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

In our equation, $a = 1$, $b = 2$, and $c = 10$. Substituting these values into the quadratic formula, we get:

$$x = \frac{-2 \pm \sqrt{2^2 - 4(1)(10)}}{2(1)}$$

$$x = \frac{-2 \pm \sqrt{4 - 40}}{2}$$

$$x = \frac{-2 \pm \sqrt{-36}}{2}$$

The discriminant, which is the term inside the square root ($b^2 - 4ac$), is $-36$. Since the discriminant is negative, the solutions will be complex numbers. This is a crucial observation, as it tells us that the equation has no real roots. Complex roots occur in conjugate pairs, meaning if $a + bi$ is a root, then $a - bi$ is also a root. Understanding the discriminant is vital in determining the nature of the roots without actually solving the equation. This saves time and provides valuable insight into the solution set. The quadratic formula is a powerful tool, but it requires careful substitution and simplification to avoid errors.

Simplifying the Complex Roots

Now, let's simplify the expression further. We have:

$$x = \frac{-2 \pm \sqrt{-36}}{2}$$

Since $\sqrt{-36} = \sqrt{36 \times -1} = \sqrt{36} \times \sqrt{-1} = 6i$, where $i$ is the imaginary unit ($\sqrt{-1}$), we can rewrite the equation as:

$$x = \frac{-2 \pm 6i}{2}$$

Now, divide both terms in the numerator by 2:

$$x = -1 \pm 3i$$

So, the two solutions are $x = -1 + 3i$ and $x = -1 - 3i$. These are complex conjugate roots, as expected. Simplifying complex roots involves understanding the properties of imaginary numbers and performing arithmetic operations with complex numbers. This step is essential to express the solutions in their simplest form. Complex numbers have real and imaginary parts, and they follow specific rules for addition, subtraction, multiplication, and division. The ability to manipulate complex numbers is crucial in various fields, including electrical engineering, quantum mechanics, and advanced mathematics.

Identifying the Correct Solution

From the simplified solutions, we can see that one of the solutions is $x = -1 - 3i$. Comparing this with the given options:

a) -12 b) -10 c) $-1 - 3i$ d) 1 e) $3i$

We can clearly see that option c) $-1 - 3i$ matches one of our solutions. The other options are incorrect. Options a), b), and d) are real numbers, but we know that the equation has complex roots. Option e) $3i$ is a purely imaginary number but does not match either of our solutions. Identifying the correct solution requires careful comparison of the calculated solutions with the given options. It also involves understanding the nature of the solutions, such as whether they are real or complex. This step is crucial to ensure that the final answer is accurate and consistent with the problem statement.

Why Other Options are Incorrect

To further reinforce understanding, let's discuss why the other options are incorrect:

• a) -12: If we substitute $x = -12$ into the original equation $x^2 + 2x = -10$, we get $(-12)^2 + 2(-12) = 144 - 24 = 120$, which is not equal to -10. Thus, -12 is not a solution.
• b) -10: Substituting $x = -10$ into the equation, we get $(-10)^2 + 2(-10) = 100 - 20 = 80$, which is also not equal to -10. Therefore, -10 is not a solution.
• d) 1: Substituting $x = 1$ into the equation, we get $(1)^2 + 2(1) = 1 + 2 = 3$, which is not equal to -10. So, 1 is not a solution.
• e) $3i$: Substituting $x = 3i$ into the equation, we get $(3i)^2 + 2(3i) = 9i^2 + 6i = -9 + 6i$, which is not equal to -10. Hence, $3i$ is not a solution.

Understanding why incorrect options are wrong is just as important as knowing the correct solution. It helps in developing a deeper understanding of the problem and the solution process. It also reinforces the importance of verifying solutions by substituting them back into the original equation. This process of elimination and verification is a valuable skill in problem-solving and critical thinking.

Conclusion

In conclusion, the solution to the quadratic equation $x^2 + 2x = -10$ is c) $-1 - 3i$. We arrived at this solution by transforming the equation into the standard quadratic form, applying the quadratic formula, simplifying the complex roots, and verifying the result. This problem highlights the importance of understanding quadratic equations, the quadratic formula, and the properties of complex numbers. By following a systematic approach and understanding the underlying concepts, we can confidently solve quadratic equations and similar mathematical problems. Mastering quadratic equations is a fundamental skill in algebra and has applications in various fields, including engineering, physics, and computer science. This step-by-step guide aims to provide a clear and thorough understanding of the solution process, enabling readers to tackle similar problems with confidence.