Class 10 Maths CFPQ Notes: Quadratic Equations | Amit Sir
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Class 10 Maths CFPQ Notes: Quadratic Equations
Competency Focused Practice Questions (CFPQ) ask you to use a concept, explain your reasoning and check what an answer means. Learn factorisation through diagrams, worked examples and original application questions with Amit Sir.
translate a situation or diagram into a quadratic equationfactorise and explain why each step worksinterpret roots and reject values that do not fit the situationspot a wrong solution and solve a fresh question independently
UnderstandSee the idea visually
ApplyFollow worked examples
PractiseWrite and ask for help
Learn, write and revise on the same note. Use the pen on a figure or example, open a blank sheet for full working, then share a doubt or solution with Amit Sir.
1. Your CFPQ confidence path
Read the situation. Choose the maths. Explain the answer.
A familiar formula can appear inside an unfamiliar story, table or diagram. Start with what is given and what you need to find. These are original CFPQ-style practice questions, not reproductions of CBSE booklet questions.
1.Read: underline the quantities, units and conditions.
2.Represent: let x be the unknown and label a sketch.
3.Reason: form an equation, solve it and explain the method.
4.Review: substitute the roots and decide which fit the context.
Confidence grows from explaining one step at a time. If you get stuck, use the hint before opening the solution.
2. The core idea
A product is zero only when at least one factor is zero.
If (x − 3)(x + 2) = 0, then x − 3 = 0 or x + 2 = 0. So x = 3 or x = −2. This is the zero-product rule. It works after the equation has been brought to one side and factorised completely.
Zero-product map
If AB = 0, A = 0 or B = 0
x − 3 = 0
x = 3
(x − 3)(x + 2) = 0
x + 2 = 0
x = −2
Always write the equation in the form ax² + bx + c = 0 before factorising.
3. Four reliable steps
Move → Factorise → Split → Check
1
Arrange
ax² + bx + c = 0
2
Factorise
Make two brackets
3
Split
Each bracket = 0
4
Check
Put roots back
1.Move every term to one side so the other side is 0.
2.Factorise the quadratic expression completely.
3.Set each factor equal to 0 and solve the two linear equations.
4.Substitute both values in the original equation to check them.
Do not divide by x or by a bracket containing x. You may accidentally lose one root.
4. Worked example: read the graph
Solve x² − 5x + 6 = 0
We need two numbers whose product is +6 and whose sum is −5: −2 and −3.
Factor-pair finder
Product
+6
Required sum
−5
Choose pair
−2, −3
Diagram: the roots are where the curve meets the x-axis
At x = 2 and x = 3, y = 0. These are the same two roots found by factorisation.
x² − 5x + 6 = 0
(x − 2)(x − 3) = 0
x − 2 = 0 or x − 3 = 0
x = 2 or x = 3
Check: 2² − 5(2) + 6 = 0 and 3² − 5(3) + 6 = 0. Both roots are valid.
5. Worked example: split the middle term
Solve 2x² + x − 6 = 0
Here ac = 2 × (−6) = −12. Use 4 and −3 because their product is −12 and their sum is +1.
When a is not 1: use the bridge
a × c = −124 and −3sum = +1
Diagram: multiply the brackets using a box
Each box multiplies its row and column labels. Adding the four terms checks (x + 2)(2x − 3). This is an algebra box; negative entries are signed terms, not areas.
2x² + 4x − 3x − 6 = 0
2x(x + 2) − 3(x + 2) = 0
(x + 2)(2x − 3) = 0
x = −2 or x = 3/2
For ax² + bx + c, split the middle term using two numbers with product ac and sum b.
6. Case-based application
Can both algebraic roots be garden dimensions?
A rectangular school garden has an area of 40 m². Its length is 3 m more than its width. Find its dimensions and explain why one root must be rejected.
Model the garden before solving
The diagram gives x(x + 3) = 40. Work out the roots, then ask which root can represent a length.
Let the width be x metres. Write the length, form an equation and find the valid dimensions. Why is the negative root unsuitable?
Need a hint?
Use area = length × width. The length is x + 3 and x must be positive. Find two numbers with product −40 and sum +3.
Check answer and reasoning
Width = x m; length = (x + 3) m.
x(x + 3) = 40 → x² + 3x − 40 = 0.
(x + 8)(x − 5) = 0 → x = −8 or x = 5.
Reject x = −8: a garden cannot have a negative width.
Width = 5 m and length = 8 m. Check: 5 × 8 = 40 m² and 8 − 5 = 3 m.
An algebraically correct root is not always a valid real-world answer. State the condition and include units.
7. Explain and correct
A student cancels x. Which solution disappears?
Riya solves x² = 5x by dividing both sides by x and writes x = 5. Is her solution complete? Explain and correct it.
Need a hint?
Move all terms to one side and take x common. Is dividing by x allowed when x might be 0?
Check answer and reasoning
Her solution is incomplete. Dividing by x assumes x ≠ 0.
x² − 5x = 0 → x(x − 5) = 0.
So x = 0 or x = 5. Both satisfy the original equation.
The missing root is 0.
In a reasoning question, give the cause of the error as well as the corrected answer.
8. Your turn
Quick independent check: show both roots
Your independent challenge
Factorise first, write both roots, then use the pen to show your method.
Solve: x² + x − 12 = 0
Need a hint?
Find two numbers with product −12 and sum +1. Then set each bracket equal to zero.
Check answer and reasoning
(x + 4)(x − 3) = 0, so x = −4 or x = 3.
Use the writing tools beside the note. Your working is saved only to your account.
9. Apply it yourself
From a fresh diagram to a justified answer
A rectangular display board has area 21 m². Its length is 4 m more than its width. Find both dimensions. Explain why you keep only one root.
Need a hint?
Let the width be x m. Use x(x + 4) = 21, then bring all terms to one side.
Check answer and reasoning
x² + 4x − 21 = 0 → (x + 7)(x − 3) = 0.
x = −7 or x = 3. Width must be positive, so reject −7.
Width = 3 m; length = 7 m.
Check: 3 × 7 = 21 m² and 7 − 3 = 4 m.
Self-check: did you define x, form the equation, solve, reject the unsuitable root and write units?
10. Compare and reason
Do two brackets always give two different roots?
A student says: “Every factorised quadratic equation has two different roots.” Use x² − 6x + 9 = 0 to decide whether this is true.
Need a hint?
Factorise the expression. Are the two brackets different?
Check answer and reasoning
x² − 6x + 9 = (x − 3)².
Both factors are x − 3, so both give x = 3.
The claim is false: this quadratic has a repeated root, 3, rather than two distinct roots.
Check: 3² − 6(3) + 9 = 0.
A counterexample can disprove an “always” claim. Explain what the example shows.
11. Ready for the next CFPQ?
Check your reasoning, not just your final number
Try the independent questions again without hints. If a step still feels unclear, write your attempt on the note and use “Ask or share solution” to discuss that exact step with Amit Sir.
1.I can define an unknown and turn a context into an equation.
2.I can explain the factor pair and the zero-product rule.
3.I can check roots and explain why a root is unsuitable.
4.I can spot a lost root or a repeated root and justify my answer.
Next step: change one number in the garden question, form the new equation and see whether your old factor pair still works.
CBSE syllabus scope
Quadratic equations in one variable: solve real roots by factorisation.
Source: Original teaching notes and practice by Amit Sir, using the competency-focused approach described in CBSE learning resources. CBSE competency-based learning framework Next teaching note: Quadratic Formula and Nature of Roots.