S5 Working With Complex Numbers Test: Performance Task 1Question 2 (2 Points)Solve The Equation: $2x^2 + X - 47 = -2$Give The Exact Solution(s), With No Rounding. If You Have Two Answers, Type Them Comma-separated From Least To Greatest. For

Solving Complex Equations: A Step-by-Step Guide to Mastering the S5 Performance Task

Introduction to Complex Numbers

Complex numbers are a fundamental concept in mathematics, and solving equations involving these numbers can be a challenging task. In this article, we will focus on solving a specific equation, $2x^2 + x - 47 = -2$, which is a part of the S5 working with complex numbers test. We will break down the solution into manageable steps, providing a clear and concise explanation of each step.

Understanding the Equation

The given equation is a quadratic equation in the form of $ax^2 + bx + c = d$. In this case, $a = 2$, $b = 1$, $c = -47$, and $d = -2$. Our goal is to find the exact solution(s) for $x$, with no rounding.

Step 1: Rearrange the Equation

To solve the equation, we need to isolate the variable $x$. We can start by rearranging the equation to get all the terms on one side:

$$2x^2 + x - 47 + 2 = 0$$

This simplifies to:

$$2x^2 + x - 45 = 0$$

Step 2: Factor the Quadratic Expression

Now that we have a quadratic equation in the form of $ax^2 + bx + c = 0$, we can try to factor the expression. Factoring a quadratic expression involves finding two numbers whose product is $ac$ and whose sum is $b$. In this case, $a = 2$, $b = 1$, and $c = -45$.

We can start by finding two numbers whose product is $2 \times -45 = -90$. These numbers are $-30$ and $3$, since $-30 \times 3 = -90$. Now, we need to check if the sum of these numbers is equal to $b$, which is $1$. Unfortunately, $-30 + 3 = -27$, which is not equal to $1$. Therefore, we cannot factor the quadratic expression using this method.

Step 3: Use the Quadratic Formula

Since we cannot factor the quadratic expression, we can use the quadratic formula to find the solutions. The quadratic formula is given by:

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

In this case, $a = 2$, $b = 1$, and $c = -45$. Plugging these values into the formula, we get:

$$x = \frac{-1 \pm \sqrt{1^2 - 4(2)(-45)}}{2(2)}$$

Simplifying the expression under the square root, we get:

$$x = \frac{-1 \pm \sqrt{361}}{4}$$

This simplifies to:

$$x = \frac{-1 \pm 19}{4}$$

Step 4: Find the Solutions

Now that we have the solutions in the form of $x = \frac{-1 \pm 19}{4}$, we can find the exact values of $x$. We have two possible solutions:

$$x_1 = \frac{-1 + 19}{4} = \frac{18}{4} = \frac{9}{2}$$

$$x_2 = \frac{-1 - 19}{4} = \frac{-20}{4} = -5$$

Therefore, the exact solutions to the equation $2x^2 + x - 47 = -2$ are $x = \frac{9}{2}$ and $x = -5$.

Conclusion

Solving complex equations can be a challenging task, but by breaking down the solution into manageable steps, we can find the exact solutions with ease. In this article, we solved the equation $2x^2 + x - 47 = -2$ using the quadratic formula. We found the exact solutions to be $x = \frac{9}{2}$ and $x = -5$. By mastering the techniques of solving complex equations, we can tackle even the most challenging problems in mathematics.

Additional Tips and Resources

• To practice solving complex equations, try using online resources such as Khan Academy or Mathway.
• For more information on quadratic equations, check out the Khan Academy video on quadratic equations.
• To learn more about complex numbers, check out the Mathway tutorial on complex numbers.

Final Thoughts

Solving complex equations is an essential skill in mathematics, and with practice and patience, anyone can master it. By following the steps outlined in this article, you can solve even the most challenging equations with ease. Remember to always check your work and to use online resources to help you practice and learn. With dedication and persistence, you can become a master of complex equations and tackle even the most challenging problems in mathematics.
Solving Complex Equations: A Q&A Guide

Introduction

Solving complex equations can be a challenging task, but with the right techniques and resources, anyone can master it. In this article, we will answer some of the most frequently asked questions about solving complex equations, providing a clear and concise explanation of each concept.

Q: What is a complex equation?

A: A complex equation is a mathematical equation that involves complex numbers, which are numbers that have both real and imaginary parts. Complex equations can be linear or nonlinear, and they can involve variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.

Q: What are the different types of complex equations?

A: There are several types of complex equations, including:

• Linear complex equations: These equations involve a linear combination of complex numbers.
• Quadratic complex equations: These equations involve a quadratic expression in the form of ax^2 + bx + c = 0.
• Polynomial complex equations: These equations involve a polynomial expression in the form of a_n x^n + a_(n-1) x^(n-1) + ... + a_0 = 0.
• Rational complex equations: These equations involve a rational expression in the form of p(x)/q(x) = 0.

Q: How do I solve a complex equation?

A: Solving a complex equation involves several steps, including:

• Simplifying the equation by combining like terms and eliminating any unnecessary variables.
• Using algebraic techniques such as factoring, expanding, and canceling to simplify the equation.
• Using numerical methods such as the quadratic formula or the Newton-Raphson method to find the solutions.
• Checking the solutions to ensure that they are valid and satisfy the original equation.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to find the solutions to a quadratic equation in the form of ax^2 + bx + c = 0. The formula is given by:

x = (-b ± √(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 and simplify the expression. You will then get two possible solutions, which are given by:

x_1 = (-b + √(b^2 - 4ac)) / 2a x_2 = (-b - √(b^2 - 4ac)) / 2a

Q: What is the difference between a complex number and a real number?

A: A complex number is a number that has both real and imaginary parts, whereas a real number is a number that has only a real part. Complex numbers are often represented in the form of a + bi, where a is the real part and b is the imaginary part.

Q: How do I add and subtract complex numbers?

A: To add and subtract complex numbers, you need to add or subtract the real parts and the imaginary parts separately. For example, if you have two complex numbers, 3 + 4i and 2 - 3i, you can add them by adding the real parts and the imaginary parts separately:

(3 + 4i) + (2 - 3i) = (3 + 2) + (4 - 3)i = 5 + i

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you need to use the distributive property and multiply the real parts and the imaginary parts separately. For example, if you have two complex numbers, 3 + 4i and 2 - 3i, you can multiply them by multiplying the real parts and the imaginary parts separately:

(3 + 4i)(2 - 3i) = (3)(2) + (3)(-3i) + (4i)(2) + (4i)(-3i) = 6 - 9i + 8i - 12i^2 = 6 - i + 12 = 18 - i

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number is a complex number that has the same real part but the opposite imaginary part. For example, if you have a complex number, 3 + 4i, its conjugate is 3 - 4i.

Q: How do I find the conjugate of a complex number?

A: To find the conjugate of a complex number, you need to change the sign of the imaginary part. For example, if you have a complex number, 3 + 4i, its conjugate is 3 - 4i.

Conclusion

Solving complex equations can be a challenging task, but with the right techniques and resources, anyone can master it. In this article, we have answered some of the most frequently asked questions about solving complex equations, providing a clear and concise explanation of each concept. We hope that this article has been helpful in your understanding of complex equations and has provided you with the tools and resources you need to solve them.