Ninth Grade · Math · week 27

Solving quadratics by factoring

Solve quadratic equations by rearranging to standard form and applying the zero product property.

The zero product property is a simple but powerful fact: if two things multiply together to make zero, at least one of them must actually BE zero - there is no other way to multiply and land on zero. Applied to a factored quadratic like (x-2)(x-3) = 0, this means either x-2 = 0 or x-3 = 0, splitting one quadratic equation into two easy linear ones.

This only works when one side of the equation is exactly 0 - not 10, not -4, exactly 0. So the first move, always, is to rearrange the equation into standard form (everything moved to one side) before factoring anything. Then factor using last term's skills, set each factor equal to zero, solve each mini-equation, and check both solutions in the ORIGINAL equation.

📋 For the grown-up teacher
Teach it (10–15 min):
  • Make the rule concrete with numbers first: "if two numbers multiply to zero, could BOTH be nonzero?" - let them convince themselves before naming the property.
  • Insist on standard form (equals zero) as a checkpoint before factoring even begins - write "= 0" and underline it as a non-negotiable first step.
  • Watch for the x² = 7x trap specifically: dividing both sides by x looks tempting and silently deletes the x = 0 solution - factoring instead keeps both.
  • Require both solutions checked in the ORIGINAL (pre-rearranged) equation, not the standard-form version, to build the full habit.
Talk about it:
  • Why must one side equal exactly zero before you can use the zero product property?
  • What solution gets lost if you divide both sides of x² = 7x by x instead of factoring?
  • How many solutions does a quadratic equation usually have, and why two instead of one?

Mastery looks like: They rearrange any quadratic to standard form, factor it, and correctly extract both solutions using the zero product property.

Common stumbles: Dividing by a variable instead of factoring (silently losing a solution); forgetting to rearrange to standard form before attempting to factor.

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