Solve Each Factor’s Zero Condition
A product of real factors equals zero only if at least one factor is zero.
About 20 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: Solve (x−2)(x+3)=0. Candidate answers: (A) x=-1 (B) x=2 only (C) x=2 or x=−3 (D) x=−2 or x=3.
- Independent check inputs: Solve (x−3)(x+4)=0. Candidate answers: (A) x=-1 (B) x=3 only (C) x=3 or x=−4 (D) x=−3 or x=4.
Learn the idea
Quadratic equations can be solved by revealing factors, making a square, or undoing a square. Each method exposes structure, and all valid roots must be retained and checked in the original equation. A product of real factors equals zero only if at least one factor is zero. This property turns a factored quadratic equation into two simpler linear equations.
Here is the method on the supplied teaching case. A product is zero only when at least one factor is zero. Solve x−2=0 and x+3=0 separately; both solutions satisfy the product equation.
The zero-product property applies when the whole product equals zero, not an arbitrary nonzero value. Keep both factor solutions, then substitute each into the original product to verify.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: Solve (x−2)(x+3)=0. Candidate answers: (A) x=-1 (B) x=2 only (C) x=2 or x=−3 (D) x=−2 or x=3. Solution: A product is zero only when at least one factor is zero. Solve x−2=0 and x+3=0 separately; both solutions satisfy the product equation. The final answer is x=2 or x=−3. The zero-product property applies when the whole product equals zero, not an arbitrary nonzero value. Keep both factor solutions, then substitute each into the original product to verify.
Check your reasoning
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Helpful tips
- A product of real factors equals zero only if at least one factor is zero.
- The zero-product property applies when the whole product equals zero, not an arbitrary nonzero value.
See the answer and explanation
x=3 or x=−4
A product is zero only when at least one factor is zero. Solve x−3=0 and x+4=0 separately; both solutions satisfy the product equation.
Take it further
Changed-input task: Solve (x−9)(x+10)=0. Candidate answers: (A) x=-1 (B) x=9 only (C) x=9 or x=−10 (D) x=−9 or x=10. Show the mathematical reason, not only the selected answer. Expected result: x=9 or x=−10. A product is zero only when at least one factor is zero. Solve x−9=0 and x+10=0 separately; both solutions satisfy the product equation.
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