Solve The Equation:$\[ X(x - 21) = 10^2 \\]

Feb 27, 2025 by ADMIN 44 views

**Solving Quadratic Equations: A Step-by-Step Guide**

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving the equation $x(x - 21) = 10^2$ . We will break down the solution into manageable steps, using a combination of algebraic manipulations and factoring techniques.

Understanding the Equation

The given equation is a quadratic equation in the form of $ax^2 + bx + c = 0$ . In this case, the equation can be rewritten as $x^2 - 21x - 100 = 0$ . Our goal is to find the values of $x$ that satisfy this equation.

Step 1: Expand and Simplify

The first step in solving the equation is to expand and simplify it. We can do this by multiplying out the brackets and combining like terms.

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the equation
equation = x*(x - 21) - 100

# Expand and simplify the equation
expanded_equation = sp.expand(equation)
simplified_equation = sp.simplify(expanded_equation)

print(simplified_equation)

This code will output the simplified equation, which is $x^2 - 21x - 100 = 0$ .

Step 2: Factor the Equation

The next step is to factor the equation. We can do this by finding two numbers whose product is $-100$ and whose sum is $-21$ . These numbers are $-25$ and $4$ , so we can write the equation as $(x - 25)(x + 4) = 0$ .

Step 3: Solve for x

Now that we have factored the equation, we can solve for $x$ by setting each factor equal to zero.

# Define the factored equation
factored_equation = (x - 25)*(x + 4)

# Solve for x
solutions = sp.solve(factored_equation, x)

print(solutions)

This code will output the solutions to the equation, which are $x = 25$ and $x = -4$ .

Conclusion

In this article, we have solved the equation $x(x - 21) = 10^2$ using a combination of algebraic manipulations and factoring techniques. We have expanded and simplified the equation, factored it, and solved for $x$ . The solutions to the equation are $x = 25$ and $x = -4$ .

Tips and Variations

To solve a quadratic equation of the form $ax^2 + bx + c = 0$ , we can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .
We can also use the factoring method to solve quadratic equations. This involves finding two numbers whose product is $c$ and whose sum is $b$ .
In some cases, we may need to use the quadratic formula to solve a quadratic equation. This involves using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .

Real-World Applications

Quadratic equations have many 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, such as sorting and searching.

Common Mistakes

Not expanding and simplifying the equation: Failing to expand and simplify the equation can lead to incorrect solutions.
Not factoring the equation: Failing to factor the equation can make it difficult to solve.
Not checking the solutions: Failing to check the solutions can lead to incorrect answers.

Conclusion

Frequently Asked Questions

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 (usually x) is two. It is typically written 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 methods to solve a quadratic equation, including factoring, using the quadratic formula, and completing the square. The method you choose will depend on the specific equation and your personal preference.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is written as x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation.

Q: What is the difference between factoring and using the quadratic formula?

A: Factoring involves expressing the quadratic equation as a product of two binomials, while using the quadratic formula involves using a mathematical formula to find the solutions. Factoring is often easier and more intuitive, but using the quadratic formula can be more efficient and accurate.

Q: How do I factor a quadratic equation?

A: To factor a quadratic equation, you need to find two numbers whose product is the constant term (c) and whose sum is the coefficient of the linear term (b). These numbers are called the factors of the quadratic equation.

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. A linear equation has only one variable and one term, while a quadratic equation has two variables and two terms.

Q: Can I use a calculator to solve a quadratic equation?

A: Yes, you can use a calculator to solve a quadratic equation. Most calculators have a built-in quadratic formula function that you can use to find the solutions.

Q: What are some common mistakes to avoid when solving a quadratic equation?

A: Some common mistakes to avoid when solving a quadratic equation include:

Not expanding and simplifying the equation
Not factoring the equation
Not checking the solutions
Using the quadratic formula incorrectly

Q: How do I check my solutions?

A: To check your solutions, you need to plug the values back into the original equation and verify that they are true. You can also use a calculator to check your solutions.

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

A: Quadratic equations have many 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, such as sorting and searching.

Q: Can I use quadratic equations to solve problems in other areas of mathematics?

A: Yes, you can use quadratic equations to solve problems in other areas of mathematics, such as algebra, geometry, and trigonometry.

Q: What are some advanced topics in quadratic equations?

A: Some advanced topics in quadratic equations include:

Quadratic equations with complex roots
Quadratic equations with irrational roots
Quadratic equations with multiple solutions
Quadratic equations with no real solutions

Q: How do I learn more about quadratic equations?

A: You can learn more about quadratic equations by:

Reading textbooks and online resources
Watching video tutorials and lectures
Practicing problems and exercises
Joining online communities and forums

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics that have many real-world applications. By understanding the basics of quadratic equations, you can solve problems in a variety of areas, including physics, engineering, and computer science. Remember to avoid common mistakes, check your solutions, and practice regularly to become proficient in solving quadratic equations.