Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 9 chapters and video availability01 · Read and understand
What you will learn
- Factor a monic quadratic and apply the zero-product property.
- Justify the method and check its domain, units, or logical conditions.
Before you start
Signed arithmetic, fraction and decimal operations, one-step equations, and introductory coordinate graphs.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Solve x²−5x+6=0 over the real numbers.
Why this math matters
Connect factored form to intercepts and solution values while checking the original equation. The example connects a stated condition to a conclusion and then checks whether the result satisfies that condition.
Set up the model
A useful answer starts with clear assumptions:
- Use the real-number system and the restrictions or data model stated in the question unless another domain is specified.
- Treat provided measurements as exact classroom-model values unless an approximation or uncertainty is stated.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Reverse a product to solve a quadratic
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Solve x²−5x+6=0 over the real numbers.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Represent the conditions
Find two numbers with sum −5 and product 6
Factoring reverses the cross-term and constant calculations.
Develop the calculation
x²−5x+6=(x−2)(x−3)
The pair −2 and −3 has the required sum and product.
Check and interpret
x=2 or x=3
A product of real numbers is zero only if at least one factor is zero; substitution verifies both roots.
The result
x=2 or x=3
A product of real numbers is zero only if at least one factor is zero; substitution verifies both roots.
Common mistakes to catch
- The zero-product rule requires the entire product to equal zero.
- Factoring over integers is not possible for every quadratic.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Solve y²+7y+12=0.
Show a hint
Find two numbers summing to seven and multiplying to twelve.
Reveal answer and explanation
y=−3 or y=−4
(y+3)(y+4)=0 gives the two solutions.
Practice 2
May you infer x=2 or x=3 from (x−2)(x−3)=10?
Show a hint
The zero-product rule needs a zero product.
Reveal answer and explanation
No
A nonzero product can come from two nonzero factors; first rewrite the equation with zero on one side and solve anew.
Take the idea with you
Connect factored form to intercepts and solution values while checking the original equation.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
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