Linear and Quadratic Equations

Arts 1–4

Foreign Languages 1–4

Solving linear and quadratic equations using factoring, completing the square, and the quadratic formula.

📘 Topic Summary

Linear and quadratic equations are fundamental concepts in algebra that can be solved using various techniques, including factoring, completing the square, and the quadratic formula. Understanding these methods is crucial for solving a wide range of mathematical problems.

📖 Glossary

Quadratic Formula: The formula x = (-b ± √(b^2 - 4ac)) / 2a used to solve quadratic equations.
Factoring: The process of expressing an algebraic expression as a product of simpler expressions, often used to solve linear and quadratic equations.
Completing the Square: A method for solving quadratic equations by manipulating the equation into the form (x + h)^2 = k.
Linear Equation: An equation in which the highest power of the variable is 1, such as 2x + 3 = 5.
Quadratic Equation: An equation in which the highest power of the variable is 2, such as x^2 + 4x + 3 = 0.

⭐ Key Points

Linear equations can be solved using addition, subtraction, multiplication, and division.
Quadratic equations can be factored into the product of two binomials.
The quadratic formula is a general method for solving all quadratic equations.
Completing the square is a useful technique for solving quadratic equations that cannot be easily factored.
Linear and quadratic equations have many real-world applications, such as modeling population growth and projectile motion.

🔍 Subtopics

Introduction to Linear Equations

A linear equation is an equation in which the highest power of the variable(s) is one. For example, 2x + 3 = 5 and x - 2 = 0 are both linear equations. The solutions to these equations can be found by isolating the variable on one side of the equation.

Factoring Quadratic Equations

Quadratic equations that can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants, can often be solved by factoring. For example, x^2 + 5x + 6 = 0 can be factored as (x + 3)(x + 2) = 0, which gives the solutions x = -3 and x = -2.

Completing the Square

Another method for solving quadratic equations is completing the square. This involves manipulating the equation to get a perfect square on one side of the equal sign. For example, x^2 + 4x + 3 = 0 can be solved by completing the square: (x + 2)^2 - 5 = 0, which gives the solutions x = -2 ± √5.

The Quadratic Formula

Quadratic equations that cannot be factored or solved by completing the square can often be solved using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. This formula will give two solutions for any quadratic equation in the form ax^2 + bx + c = 0.

Solving Systems of Linear Equations

Systems of linear equations are sets of two or more equations that must be true simultaneously. These systems can be solved using a variety of methods, including substitution and elimination. For example, the system x + y = 2 and 2x - 2y = -1 can be solved by substituting one equation into the other: (2x - 2y) + 2y = -1 + 2y = 4, which gives the solution x = 1 and y = 1.

Quadratic Equation Applications

Quadratic equations have many real-world applications. For example, projectile motion can be modeled using quadratic equations, as can the trajectory of a thrown ball or the path of a satellite in orbit around the Earth.

Common Mistakes in Solving Quadratic Equations

When solving quadratic equations, it is easy to make mistakes. Some common errors include forgetting to check for extraneous solutions, not simplifying the equation properly, and using the wrong method to solve the equation.

Practice Problems and Exercises

Solve the following quadratic equations: x^2 + 4x + 3 = 0, x^2 - 5x + 6 = 0, and x^2 + 2x - 3 = 0. Check your answers using the quadratic formula.

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