Eleventh Grade · Math · week 9

Solving quadratics by factoring

Solve quadratic equations by factoring into two binomials and applying the zero product property.

The zero product property is delightfully simple: if two numbers multiply to give 0, at least one of them MUST be 0 -- there is no other way to multiply and land on zero. That single fact turns a factored quadratic equation into two easy linear equations. The entire skill this week is factoring the quadratic first so that property can even apply.

For x^2 + bx + c, hunt for two numbers that multiply to c and add to b -- those numbers become the constants in your two binomial factors. For a leading coefficient other than 1 (like 2x^2+5x-3), the same idea works but you multiply a and c first, split the middle term using that pair, and factor by grouping. Either way, the equation must be set equal to 0 BEFORE you can use the zero product property -- factoring a quadratic that equals some other number does not tell you anything useful yet.

📋 For the teacher
Teach it (10–15 min):
  • Drill the number-pair search as its own mini-skill first, with no equation attached: "give me two numbers that multiply to -20 and add to -1" -- fluency here makes everything else fast.
  • Say the zero product property out loud as a full sentence every time: "if this times that equals 0, then this is 0 OR that is 0." The logic, not just the mechanics, needs to stick.
  • For a leading coefficient that is not 1, walk through the multiply-a-and-c-then-group method slowly the first few times -- it feels like a new trick but is the same number hunt underneath.
  • Always require the equation to be set equal to 0 first -- if they factor an equation that equals 12, stop and ask what property applies now (none does yet).
Talk about it:
  • Why does the zero product property only work when one side of the equation is 0?
  • How do you find the two numbers you need when the leading coefficient of x^2 is 1?
  • What changes about the number hunt when the leading coefficient is not 1?

Mastery looks like: They factor a quadratic (leading coefficient 1 or otherwise), set it to 0, and apply the zero product property to find both solutions.

Common stumbles: Trying to use the zero product property before the equation equals 0; finding a factor pair that multiplies correctly but does not add correctly (or vice versa).

See it visually

Interactive, spoken-aloud tools that show this idea in motion.

MultiplicationEqual groups and arrays make products finally click.try →
Homework for this lessonType it, write it with a stylus, or print it for pencil & paper.open →
Records attendance + 15 min of Math for this learner.
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