Quadratic Expression Problems

A quadratic expression is a polynomial of degree 2, written in the general form ax² + bx + c, where a, b, and c are constants and a ≠ 0. Unlike quadratic equations, expressions do not include an equals sign—they represent a value rather than something to work through. Examples include 3x² + 2x − 5 and x² − 9. Understanding quadratics at this level is essential before you move on to quadratic equations and the quadratic formula.

A quadratic expression is simply a polynomial like ax² + bx + c with no equals sign. A quadratic equation sets that expression to zero: ax² + bx + c = 0. You simplify or rewrite expressions, but you work through equations to get specific values of x. Quadratic equations can yield two answers, one answer, or no real answers depending on the discriminant. Many problem types in quadratics require you to first simplify an expression and then tackle the resulting equations.

To rewrite the expression as a product of two binomials: when a = 1, look for two numbers that multiply to c and add to b. When a ≠ 1, use the AC method: multiply a × c, locate a pair of numbers that adds to b, split the middle term, and group. Always check for special patterns like difference of squares (a² − b²) or perfect trinomials first. Once done, you can tackle quadratic equations by setting each binomial to zero, giving you the answers for x.

The quadratic formula is x = (−b ± √(b² − 4ac)) / 2a. Use it whenever you need to handle quadratic equations that are hard to rewrite as binomials. It works for all quadratic equations—just substitute the values of a, b, and c, then compute the discriminant. If the discriminant is positive you get two answers; if it is zero you get one; and if it is negative there are no real answers. The quadratic formula is one of the most widely used tools in all of quadratics, so practice applying it to different problem types until equations become second nature.

There are three main forms used across quadratics: Standard form (ax² + bx + c), which is the most common and useful for applying the quadratic formula. Factored form, such as a(x − r)(x − s), which reveals the roots of the corresponding quadratic equations. Vertex form, a(x − h)² + k, which shows the minimum or maximum value k and is useful for graphing. Converting between these forms is a core skill in quadratics—for example, you can transform standard form into vertex form, and setting the factored form to zero lets you get the answers directly.

Quadratic word problems often ask you to determine the dimensions of a rectangle or garden. A typical problem might say a garden is 5 x meters in length and some number of meters wide, and you need to get the length and width given the area. Start by assigning x to one dimension—say, the width. Express the length in terms of x using information from the problem. Multiply length × width to write an area expression, set it to the given area, and solve the resulting quadratic equations. For example, if a rectangle is x meters wide and (x + 4) meters in length with an area of 60 m², the problem leaves you with x² + 4x = 60. Solve to find which answer yields valid dimensions—the width, length, and sides must all be positive numbers. These word problems also appear with fences, garden borders, and outdoor sections where you find the sides of shapes.

Revenue and cost word problems model profit as a quadratic expression. A typical problem gives a price and demand relationship, and you need to determine the values that maximize or minimize revenue. Write the revenue expression as price × quantity, expand it, and you get a quadratic in standard form. To locate the maximum revenue, use the quadratic formula or convert to vertex form. The problem usually asks for specific numbers—like the price that maximizes revenue. Always check your result by substituting values back into the equations. These types of problem appear across applied quadratics.

You will discover a wide range of problem types covering all of quadratics. The section on expression basics covers identifying and evaluating expressions. The topic on factoring techniques includes every method from simple trinomials to difference of squares. Advanced problem sets focus on comparing equivalent forms. Word problems include area and dimensions problems (find the length, width, and sides of a rectangle or garden), revenue and cost models, and pattern problems with numbers. Each problem comes with a step-by-step answer and a video explanation so you can check your work. In total, you can solve over 160 problems to build lasting fluency in quadratics.

It depends on your starting level, but most students who solve problems consistently notice improvement within a few weeks. We recommend at least 5 x problems per day in the area of quadratics that challenges you most. Check every answer carefully and redo any problem you got wrong. Over time, equations become easier to handle, the quadratic formula becomes second nature, and word problems no longer feel intimidating. Watch the video walkthroughs for any problem where the answer surprised you—understanding why you made a mistake creates a much stronger impression than simply moving on.

Quadratic expressions are foundational in algebra and appear throughout mathematics, science, and real-world applications. They are used to model projectile motion, calculate area for rectangles and other shapes, optimize revenue and profit, and determine the dimensions of structures. Mastering how to solve quadratics is essential for working with quadratic equations, graphing parabolas, and tackling advanced math. Each topic builds on the last, so the earlier you start on these problem types, the easier the rest of quadratics becomes.