A reliable method
Set an equation equal to zero before factoring. For ax² + bx + c = 0, the roots sum to −b/a.
The common trap
Setting factors equal to zero works only when their product equals zero. A vertex value and a root are different features.
Make the connection
The discriminant b² − 4ac determines the number of distinct real roots: positive means two, zero means one, and negative means none.
Select a form for the requested feature
For y=(x−3)²−4, the vertex is (3,−4), so the minimum output is −4. To find zeros, set y=0: (x−3)²=4, giving x=1 or 5. Expanding gives x²−6x+5; the roots sum to 6 and multiply to 5. Each form reveals a different feature of the same curve.
Keep zero-product reasoning conditional
From (x−1)(x−5)=0, either factor may be zero. From (x−1)(x−5)=8, setting a factor to zero does not solve the equation because the product would then be zero. Move all terms to one side before applying the zero-product property. A graph can suggest roots, but substitution confirms exact values.
Try, then check
For 2x²−8x+c=0 to have one distinct real root, set the discriminant 64−8c to zero. Thus c=8 and the equation becomes 2(x−2)²=0. Check that the repeated root is 2. The constant 8 is the answer to the parameter question; the root 2 answers a different question.
What is the larger solution of (x − 3)(x + 4) = 0?
- A product is zero if at least one factor is zero.
- The roots are x = 3 and x = −4.
- The larger root is 3.
Compare the other answer choices
Lost the sign when solving x+b=0.
Lost the sign when solving x−a=0.
Reported the sum of the roots.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
Official mathematics skillsSiyavula Grade 10 mathematicsQuadratic equations by factoring