Intro Solving Quadratics — I can solve simple quadratic equations by factoring when solutions are integers at an introductory level.

Humans have been solving equations with squared unknowns for thousands of years. Ancient civilizations needed these calculations for surprisingly practical reasons — measuring land, designing buildings, and calculating the paths of projectiles. The idea of a quadratic equation — an equation where the variable is raised to the second power — appeared long before anyone used modern algebraic symbols. Over centuries, mathematicians developed increasingly elegant methods to solve them, and factoring became one of the most intuitive approaches for equations with integer solutions.

So here's the central question this lesson tackles: when you see an equation like x² − 5x + 6 = 0, how do you find the values of x that make it true? Factoring gives you a powerful, fast method — especially when those solutions turn out to be nice whole numbers. Let's learn how.

Before you can factor a quadratic, you need to understand a few building blocks. A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The highest power of the variable is 2, which is what makes it "quadratic" (from the Latin quadratus, meaning "square"). In this introductory lesson, we focus on equations where a = 1, so the leading coefficient is simply 1.

The diagram below illustrates the core process of factoring a quadratic like x² + bx + c. Your job is to find two integers, p and q, that satisfy two conditions simultaneously: they must multiply to give c and add to give b. When both conditions are met, you've cracked the factorization.

Notice how the entire method hinges on two simple questions: "What multiplies to c?" and "What adds to b?" If you can find a pair of integers that answers both questions, the rest is just algebra. In the next section, we'll formalize the math behind each step.

Let's pin down the algebra. When you multiply two binomials using the FOIL method (First, Outer, Inner, Last), you can see exactly why the sum-and-product trick works. The expansion (x + p)(x + q) = x² + qx + px + pq = x² + (p + q)x + pq shows that the coefficient of x is the sum p + q, and the constant term is the product pq.

The trickiest part of factoring is finding the right pair of integers. A systematic approach makes this much easier. Start by listing all factor pairs of c (the constant term), including negative pairs when c is positive and mixed-sign pairs when c is negative. Then check which pair adds up to b. The diagram below walks through this process for x² − 5x + 6 = 0.

Here are a few sign-pattern tips. When c is positive, both p and q share the same sign — both positive if b is positive, both negative if b is negative. When c is negative, one of p and q is positive and the other is negative. Keeping these patterns in mind cuts your search time in half.

Let's solve x² + 3x − 10 = 0 by factoring, walking through each step carefully.

Factoring is a fast and elegant method, but it has its sweet spot. Understanding where it shines and where it struggles will help you choose the right tool for each problem.

Factoring is your first tool for solving quadratics, but it won't work for every equation. As you progress in algebra, you'll encounter quadratics with non-integer or irrational solutions — like x² − 3x + 1 = 0 — where factoring over the integers is impossible. That's when you'll learn more powerful methods.

Even after you learn these advanced techniques, factoring remains valuable. It builds the intuition you need for completing the square, and the Zero Product Property shows up again in higher-level math when you solve polynomial equations of degree 3 and beyond. Mastering factoring now sets a strong foundation for everything that follows.