Algebraic Identities, Equations and Inequalities

Arts 1–4

Foreign Languages 1–4

Includes solving linear, quadratic, and cubic equations and inequalities.

📘 Topic Summary

Algebraic identities, equations and inequalities are fundamental concepts in Additional Mathematics that enable students to solve various types of problems. This study guide will provide a comprehensive overview of these topics, including solving linear, quadratic, and cubic equations and inequalities.

📖 Glossary

Linear Equation: An equation in which the highest power of the variable is 1.
Quadratic Equation: An equation in which the highest power of the variable is 2.
Cubic Equation: An equation in which the highest power of the variable is 3.
Inequality: A statement that compares two expressions using greater than, less than, or equal to symbols.

⭐ Key Points

Linear equations can be solved by adding or subtracting the same value to both sides.
Quadratic equations can be factored into the product of two binomials.
Cubic equations can be solved using various methods such as factoring, quadratic formula, and numerical methods.
Inequalities can be solved by applying basic algebraic operations and using properties of inequalities.

🔍 Subtopics

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. To solve a linear equation, add or subtract the same value to both sides, and then multiply or divide by the coefficient of the variable. This can be done using inverse operations.

Quadratic Equations

A quadratic equation is an equation in which the highest power of the variable(s) is two. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants. To solve a quadratic equation, use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.

Cubic Equations

A cubic equation is an equation in which the highest power of the variable(s) is three. The general form of a cubic equation is ax^3 + bx^2 + cx + d = 0, where a, b, c, and d are constants. Cubic equations can be solved using various methods, including factoring, the rational root theorem, and numerical methods.

Inequalities

An inequality is an equation with a less-than or greater-than symbol instead of an equal sign. For example, 2x + 3 > 5 and x - 2 ≤ 0 are both inequalities. To solve an inequality, perform the same operations on both sides as you would for an equality, but remember to reverse the direction of the inequality symbol if you multiply or divide by a negative number.

Systems of Equations

A system of equations is a set of two or more equations that must be true simultaneously. Systems can be solved using various methods, including substitution, elimination, and matrices. The goal is to find the values of the variables that satisfy all the equations in the system.

Polynomial Identities

A polynomial identity is an equation involving polynomials that holds true for all values of the variable(s). For example, (x + y)^2 = x^2 + 2xy + y^2 and (x - y)^2 = x^2 - 2xy + y^2 are both polynomial identities. These identities can be used to simplify expressions and solve equations.

Solving Quadratic Inequalities

Quadratic inequalities can be solved by using the same methods as quadratic equations, but with an inequality symbol instead of an equal sign. The solutions are the values of the variable that satisfy the inequality. For example, solving x^2 + 4x - 3 > 0 involves finding all values of x that make the expression true.

Cubic Inequalities

Cubic inequalities can be solved by using numerical methods or approximations, as there is no general formula to solve them. The solutions are the values of the variable that satisfy the inequality. For example, solving x^3 + 2x^2 - 5x - 6 ≥ 0 involves finding all values of x that make the expression true.

Graphing Linear and Quadratic Functions

Linear functions have a constant rate of change, while quadratic functions have a changing rate of change. The graph of a linear function is a straight line, while the graph of a quadratic function is a parabola. The x-intercepts are the points where the function crosses the x-axis, and the y-intercept is the point where the function crosses the y-axis.

Modeling Real-World Problems

Algebraic identities, equations, and inequalities can be used to model real-world problems involving growth, decay, and optimization. For example, a company's profit can be modeled using a quadratic equation, while the spread of a disease can be modeled using an exponential function.

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