What Are The Roots Of The Equation $x^2 + 10x + 50 = 0$ In Simplest $a + Bi$ Form?

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Introduction to Quadratic Equations

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

The Quadratic Formula

To find the roots of a quadratic equation, we can use the quadratic formula, which is given by:

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

This formula is derived from the fact that the quadratic equation can be factored as $(x - r)(x - s) = 0$, where $r$ and $s$ are the roots of the equation. By expanding the product and equating the coefficients, we get the quadratic formula.

Applying the Quadratic Formula to the Given Equation

Now, let's apply the quadratic formula to the given equation $x^2 + 10x + 50 = 0$. We have $a = 1$, $b = 10$, and $c = 50$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-10 \pm \sqrt{10^2 - 4(1)(50)}}{2(1)}$$

Simplifying the Expression Under the Square Root

To simplify the expression under the square root, we need to calculate the value of $10^2 - 4(1)(50)$. This is equal to $100 - 200 = -100$. Therefore, the expression under the square root is $-100$.

Simplifying the Quadratic Formula

Now that we have simplified the expression under the square root, we can simplify the quadratic formula. We get:

$$x = \frac{-10 \pm \sqrt{-100}}{2}$$

Simplifying the Square Root of a Negative Number

The square root of a negative number is an imaginary number, which can be written in the form $bi$, where $b$ is a real number and $i$ is the imaginary unit, which satisfies $i^2 = -1$. Therefore, we can write $\sqrt{-100}$ as $10i$.

Simplifying the Quadratic Formula Further

Now that we have simplified the square root of the negative number, we can simplify the quadratic formula further. We get:

$$x = \frac{-10 \pm 10i}{2}$$

Simplifying the Expression Further

To simplify the expression further, we can divide both the real and imaginary parts by 2. This gives us:

$$x = -5 \pm 5i$$

Conclusion

In this article, we have found the roots of the equation $x^2 + 10x + 50 = 0$ in simplest $a + bi$ form. The roots are $-5 + 5i$ and $-5 - 5i$. These roots are complex numbers, which are of the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

The Importance of Complex Numbers

Complex numbers are an essential concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. They are used to represent quantities that have both real and imaginary parts, and they are used to solve equations that have no real solutions.

The Applications of Complex Numbers

Complex numbers have numerous applications in various fields, including:

• Physics: Complex numbers are used to represent the amplitude and phase of waves, and they are used to solve equations that describe the behavior of physical systems.
• Engineering: Complex numbers are used to represent the impedance and admittance of electrical circuits, and they are used to solve equations that describe the behavior of electrical systems.
• Economics: Complex numbers are used to represent the growth rate and interest rate of economic systems, and they are used to solve equations that describe the behavior of economic systems.

The Future of Complex Numbers

Complex numbers are an essential concept in mathematics, and they will continue to play a vital role in the development of mathematics and science. As technology advances, complex numbers will be used to solve increasingly complex problems, and they will be used to develop new mathematical models and theories.

The Limitations of Complex Numbers

While complex numbers are a powerful tool for solving equations, they have some limitations. For example, they are not suitable for solving equations that have no real solutions, and they are not suitable for solving equations that have a large number of variables.

Conclusion

In this article, we have found the roots of the equation $x^2 + 10x + 50 = 0$ in simplest $a + bi$ form. The roots are $-5 + 5i$ and $-5 - 5i$. These roots are complex numbers, which are of the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit. Complex numbers are an essential concept in mathematics, and they have numerous applications in various fields. They will continue to play a vital role in the development of mathematics and science, and they will be used to solve increasingly complex problems.

Introduction

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. In this article, we will answer some frequently asked questions about complex numbers.

Q: What is a complex number?

A: A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, which satisfies $i^2 = -1$.

Q: What is the imaginary unit?

A: The imaginary unit, denoted by $i$, is a mathematical concept that satisfies $i^2 = -1$. It is used to extend the real number system to the complex number system.

Q: How do I add complex numbers?

A: To add complex numbers, you simply add the real parts and the imaginary parts separately. For example, if you have two complex numbers $a + bi$ and $c + di$, their sum is $(a + c) + (b + d)i$.

Q: How do I subtract complex numbers?

A: To subtract complex numbers, you simply subtract the real parts and the imaginary parts separately. For example, if you have two complex numbers $a + bi$ and $c + di$, their difference is $(a - c) + (b - d)i$.

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you can use the distributive property and the fact that $i^2 = -1$. For example, if you have two complex numbers $a + bi$ and $c + di$, their product is $(ac - bd) + (ad + bc)i$.

Q: How do I divide complex numbers?

A: To divide complex numbers, you can use the fact that $i^2 = -1$ and the distributive property. For example, if you have two complex numbers $a + bi$ and $c + di$, their quotient is $\frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$.

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number $a + bi$ is $a - bi$. The conjugate of a complex number is used to simplify expressions and to eliminate the imaginary part.

Q: How do I simplify complex expressions?

A: To simplify complex expressions, you can use the fact that $i^2 = -1$ and the distributive property. You can also use the conjugate of a complex number to eliminate the imaginary part.

Q: What are the applications of complex numbers?

A: Complex numbers have numerous applications in various fields, including physics, engineering, and economics. They are used to represent quantities that have both real and imaginary parts, and they are used to solve equations that have no real solutions.

Q: What are some common mistakes to avoid when working with complex numbers?

A: Some common mistakes to avoid when working with complex numbers include:

• Not using the distributive property: When multiplying complex numbers, it's essential to use the distributive property to simplify the expression.
• Not using the conjugate: When simplifying complex expressions, it's essential to use the conjugate to eliminate the imaginary part.
• Not checking the signs: When working with complex numbers, it's essential to check the signs of the real and imaginary parts to ensure that the expression is correct.

Conclusion

In this article, we have answered some frequently asked questions about complex numbers. Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields. By understanding complex numbers and how to work with them, you can solve a wide range of problems and make new discoveries.

Additional Resources

If you want to learn more about complex numbers, here are some additional resources:

• Textbooks: There are many textbooks available on complex numbers, including "Complex Analysis" by Serge Lang and "Complex Numbers and Geometry" by David A. Brannan.
• Online resources: There are many online resources available on complex numbers, including Khan Academy, MIT OpenCourseWare, and Wolfram Alpha.
• Software: There are many software packages available that can help you work with complex numbers, including Mathematica, Maple, and MATLAB.