Curriculum

Factoring Lessons

Learn factoring from the fundamentals through quadratics, discriminants, functions, and inequalities with step-by-step explanations, worked examples, and interactive practice.

01

Factoring Fundamentals

What factors are, common factors, and factoring expressions completely.

What Is a Factor? A factor is a number or expression that is multiplied by another to make a product. Understanding factors is the first step toward factoring algebraic expressions and quadratics. Common Factors Common factors are factors shared by two or more numbers. Finding them helps you simplify numbers and identify what can be taken outside an algebraic expression. Greatest Common Factor (GCF) The greatest common factor, or GCF, is the largest factor shared by two or more numbers. It tells you the biggest number you can divide out of every term. Factoring Out the GCF Factoring out the GCF rewrites a sum as multiplication. You place the greatest factor shared by every term outside parentheses, then divide each term by it. Factoring Completely To factor completely means to continue until none of the factors can be factored any further using the method you are learning.

02

Factoring Quadratics

Turn expansion around: find the two numbers, then write the brackets.

Factoring as Reverse Expansion Expanding multiplies brackets to create an expression. Factoring works backward: it starts with the expression and recovers the brackets that produced it. Finding Factor Pairs for Quadratics For a quadratic in the form x² + bx + c, the key step is finding two numbers whose product is c and whose sum is b. Factoring Positive Quadratics When x² + bx + c has positive b and c, both numbers in the factor pair are positive. This makes it a useful first pattern to master. Factoring Quadratics with Negative Terms If the constant is positive but the middle coefficient is negative, the two numbers in the brackets must both be negative. Factoring Quadratics with Mixed Signs A negative constant means the two numbers in the factor pair have opposite signs. Their difference determines the middle coefficient. How to Factor Quadratics Factoring a quadratic means rewriting it as a product of simpler expressions. For x² + bx + c, you are looking for two numbers that multiply to c and add to b.

03

Advanced Factoring

Difference of squares and factoring with a leading coefficient.

Difference of Squares A difference of squares is a special factoring pattern: one perfect square minus another perfect square. Factoring Quadratics with a Leading Coefficient When the coefficient of x² is not 1, both the first and last terms influence the factor pair. One reliable approach is to multiply a × c and split the middle term.

04

Solving Quadratics

Once factored, each bracket gives a root.

Solving Quadratics by Factoring Once a quadratic equation is factored and set equal to zero, each factor can be used to find a solution, also called a root.

05

The Discriminant

Meet Δ, calculate it, and use it to predict roots — then parameters.

The Quadratic Discriminant The discriminant is the expression b² − 4ac inside the quadratic formula. Its value tells you how many real roots a quadratic equation has. Calculating the Discriminant Calculating the discriminant is a substitution problem, but signs and the order of operations matter. Square b before subtracting 4ac. What Does the Discriminant Mean? The sign of the discriminant tells you how many real roots a quadratic equation has before you solve it. Predicting Roots with the Discriminant You can predict the number of real roots without fully solving the quadratic. Identify the coefficients, calculate Δ, and interpret its sign. Working Backward from the Number of Roots The discriminant rule works in both directions. If you know how many real roots an equation has, you can state what must be true about Δ. The Discriminant with Parameters A parameter is a letter inside a coefficient. Instead of producing one fixed discriminant value, substitution produces an expression in that parameter. Discriminant Parameter Inequalities Parameter problems combine the discriminant rule with an equation or inequality. The requested number of roots determines whether Δ must be positive, zero, or negative.

06

Quadratic Inequalities

Sketch roots, then read the interval from the parabola’s sign.

Quadratic Inequalities and Number Lines A quadratic inequality asks where a quadratic expression is positive, negative, or equal to zero. Its roots divide the number line into intervals that you can test.

07

Parameter Problems

Find values of k that give two roots, one root, or none.

Finding Values of k for Real Roots Questions about values of k are often discriminant problems. Build Δ in terms of k, then apply the condition for the requested number of roots.

08

Functions & Inverses

Function notation, evaluating f(x), and finding inverse functions.

What Is a Function? A function is a rule that takes an input and produces exactly one output. That single-output rule is the foundation for function notation and inverse functions. Function Notation Function notation names a rule and shows its input. The expression f(x) means the output of f when the input is x. Evaluating Functions Evaluating a function means finding the output for a chosen input. Replace every copy of the input variable with the given number, then simplify. What Is an Inverse Function? An inverse function undoes the original function. If f sends an input to an output, f⁻¹ sends that output back to the original input. Finding Inverse Functions To find an inverse algebraically, write y = f(x), swap x and y, then solve for y. The resulting formula is f⁻¹(x). Checking Inverse Functions Two functions are inverses when each undoes the other. The strongest algebraic check is composition: f(f⁻¹(x)) and f⁻¹(f(x)) should both simplify to x.