Factoring Quadratics

What to say / Spoken Script

Hello everyone! Today we are diving into quadratic expressions of the form ax² + bx + c. The word 'quadratic' comes from the Latin word 'quadratus' meaning square, because the highest power of our variable x is 2. The values a, b, and c are coefficients: a is the coefficient of x², b is the coefficient of x, and c is the constant term. Factoring a quadratic means writing it as the product of two simpler linear expressions. Geometrically, this is like finding the width and height of a rectangle with a total area equal to ax² + bx + c.

What to do / Action Guide

Direct students' attention to the equation ax² + bx + c and show them the corresponding quadratic graph on the screen.

Pedagogical Tips

Ask students: If we have the expression x² - 4, what is the value of b? (b = 0). Remind them that if no number is written in front of x², the coefficient a is 1.

What to say / Spoken Script

Let's start with monic quadratics where a = 1. E.g., x² + 5x + 6 (a = 1, b = 5, c = 6). We want to factor this into (x + p)(x + q). When expanded, this is x² + (p+q)x + pq. Notice that the two mystery numbers p and q must multiply to get c (6) and add up to get b (5). Let's check the factors of 6: 1 × 6 = 6 (but 1 + 6 = 7), and 2 × 3 = 6 (with 2 + 3 = 5). So the numbers are 2 and 3. The factored form is (x + 2)(x + 3). Geometrically, as shown on the right, we arrange algebra tiles of x², five x-rectangles, and six 1-squares to form a clean rectangle of width x + 2 and height x + 3.

What to do / Action Guide

Explain the factoring equation on the board, then point out how the algebra tiles grid fits perfectly into a large rectangle of height x + 3 and width x + 2.

Pedagogical Tips

Emphasize to students: always double check your factors by multiplying them back using expansion (FOIL method) to verify you recover the original expression.

What to say / Spoken Script

Let's factor a non-monic quadratic where a > 1 and c > 1. E.g., 2x² + 7x + 3 (a = 2 > 1, b = 7, c = 3 > 1). We use the Grouping Method. Step 1: Multiply a by c: 2 × 3 = 6. Step 2: Find two numbers that multiply to 6 and add to b (7). The numbers are 6 and 1. Step 3: Rewrite the middle term 7x using these numbers: 2x² + 6x + x + 3. Step 4: Group the terms: (2x² + 6x) + (x + 3). Step 5: Factor out GCF from each group: 2x(x + 3) + 1(x + 3). Finally, factor out the common binomial (x + 3) to get (2x + 1)(x + 3). Geometrically on the right, we arrange two x² tiles, seven x-rectangles, and three 1-squares to form a rectangle of width 2x + 1 and height x + 3.

What to do / Action Guide

Walk students through the grouping algebra steps on the blackboard. Show them how the factoring steps match the groupings inside the algebra tile layout.

Pedagogical Tips

Remind students to always write the '+ 1' explicitly in the grouping step so they do not lose it when factoring out the common bracket.

What to say / Spoken Script

To solve any quadratic trinomial ax² + bx + c: First, look for a Greatest Common Factor (GCF) and pull it out. Second, identify coefficients a, b, and c, and calculate product a × c. Find two numbers that multiply to a × c and add to b. Third, rewrite the middle term bx using these two numbers, group the terms in pairs, factor out GCF of each pair, and rewrite as product of two binomials. Let's study this flowchart carefully before trying a real-world problem.

What to do / Action Guide

Explain the general factoring flowchart steps, walking students through the sequence of GCF, Finding Factors, and Grouping.

Pedagogical Tips

Learning this flowchart makes factoring any quadratic automatic! Practice identifying a, b, and c immediately for every problem.

What to say / Spoken Script

Let's apply our knowledge to a real-world problem. Suppose a farmer has a rectangular field with a total area given by the quadratic expression x² + 8x + 15. By factoring this quadratic, we can find the algebraic expressions for the width and height of the field! Since a = 1, b = 8, and c = 15, we need two numbers that multiply to 15 and add to 8. The numbers are 3 and 5. This means the field has a width of x + 5 and a height of x + 3. Factoring helps us solve real-world geometry, architectural design, and land optimization problems!

What to do / Action Guide

Explain how factoring relates a quadratic area back to linear dimensions. Point to the fence design diagram on the screen.

Pedagogical Tips

Use this example to show students how algebra is used in engineering, architecture, and landscaping to model physical dimensions.