CBSE Class 10 · Mathematics
Quadratic Equations
Official NCERT chapter from Mathematics (book code jemh1). ExamMaster notes are original teaching at CBSE Class 10 depth.
This lesson follows the official chapter “Quadratic Equations” in Mathematics. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.
- CBSE Class 10
- Medium level
- 8 concepts
1Quadratic Equations
A quadratic equation is ax² + bx + c = 0 with a ≠ 0. x² − 5x + 6 = 0 is one; 3x − 5 = 0 is linear, not this. The unknown is x; the writing is degree 2. A pretty parabola sketch is not the equation until it is set to 0.
y = x² − 5x + 6 is a curve; the equation asks where that curve is zero.
Figure. y = x^2 - 4 meets the x-axis at two places. Those meetings are the real roots of x^2 - 4 = 0.
How it works
- See the highest power 2 and a ≠ 0The sort.
- Set the writing = 0 if it was a y = …The equation.
- Name x as the unknownQuadratic.
2Solution of a Quadratic Equation by Factorisation
Factorisation: write ax²+bx+c as a product of linear factors and set each factor to 0. x² − 5x + 6 = (x−2)(x−3), so x = 2 or x = 3. A pair that multiplies to c and adds to b (when a = 1) is the hunt; an expand-check is required.
Writing any two brackets that look busy is not the hunt.
Figure. A product is zero only when a factor is zero. Each linear factor names one root.
How it works
- Hunt two numbers (product c, sum b when a = 1)The pair.
- Write the bracketsThe candidate.
- Expand, then set each factor = 0The solutions.
x² − 5x + 6 = 0
Solve by factorisation.
- Pair−2 and −3 — product 6, sum −5
- Factors(x−2)(x−3)=0
- x2 or 3
Pro tip. Expand-check, then each factor zero.
3Nature of Roots
Nature of roots uses D = b² − 4ac (as taught): D > 0 two distinct real, D = 0 one repeated real, D < 0 no real root in this class’s drawer. For x² − 5x + 6, D = 25 − 24 = 1 > 0. Nature is the sort, not the values — values need factor or the formula if taught.
Calling D “the answer” without saying two/one/none is a heading-miss.
Figure. Read D = b^2 - 4ac before solving. Sign of D is the number of real meetings with the x-axis.
How it works
- Compute D = b² − 4acThe discriminant.
- Sort two distinct / repeated / no realThe nature.
- Then find the values only if askedNature first if that was the heading.
4A definition is a test you can run
Quadratic is a test: degree 2, set to 0, a ≠ 0. If you cannot name the degree, you have a writing only.
A pretty x² banner that is never set to 0 is not the equation.
Figure. A number is a root only when substituting it makes the quadratic equal zero. The test is the substitution, not the look of the letter.
How it works
- Name the degree and a ≠ 0The object.
- See = 0The test.
- Then the word has contentThe definition ran.
5Name the given before the unknown
The given is a, b, c. The unknown is x or the nature. Copy the signs before you hunt −2 and −3.
Using +5x as if b were +5 in x²−5x+6 is a silent given.
Figure. Name a, b and c from the written quadratic before you hunt for x. The unknown is last.
How it works
- Copy a, b, cThe given.
- Name roots or D-sortThe unknown.
- Then factor or compute DGiven first.
6One worked case is enough at this class
One quadratic is enough. A page of the same 2-or-3 clone does not add a new D.
Ten twins of D = 1 are still one idea.
Figure. One fully written case is the pattern: factor, set each factor to zero, name both roots.
How it works
- Solve or sort one equationThe case.
- A second only as a checkOptional.
- StopThis class.
7A miss: swapping the school name for the picture
A fountain-arc photo is a setting. The maths is ax²+bx+c = 0. Water is not a root.
A bridge-arch is not D.
Figure. The school name “quadratic” is a label. The picture that earns the name is the graph meeting the x-axis.
How it works
- Keep the photo as a storyA setting.
- Write the equation and factor or sort DThe maths.
- Then x or the natureDo not swap.
8Check by the opposite action or the opposite test
Check roots by substituting back into ax²+bx+c. Check nature by recomputing D. p(2)=0 and p(3)=0 must hold for x²−5x+6.
A factor-pair that fails the expand is not a check.
Figure. A claimed root is honest only when putting it back yields 0. The opposite of solving is substituting.
How it works
- State 2 or 3, or “two distinct”The claim.
- Substitute or recompute DThe check.
- Mend if they disagreeHonest.
For x² − 5x + 6, D is
- 1 — two distinct real roots
- 0
- −1
25 − 24 = 1.
Notes
- The official chapter title is “Quadratic Equations”. Teach the school test for that title, not a contest shortcut.
- If a step needs a later class, stop. The next official chapter will pick it up.
Formulas
- D = b² − 4ac
- D>0 two distinct real; D=0 repeated; D<0 no real (this class)
Recap
Hold these pegs from the official chapter “Quadratic Equations”. The wording is ExamMaster’s teaching, not a textbook recap.
- Quadratic Equations
- A quadratic equation is ax² + bx + c = 0 with a ≠ 0.
- Solution of a Quadratic Equation by Factorisation
- Factorisation: write ax²+bx+c as a product of linear factors and set each factor to 0.
- Nature of Roots
- Nature of roots uses D = b² − 4ac (as taught): D > 0 two distinct real, D = 0 one repeated real, D < 0 no real root in this class’s drawer.
- A definition is a test you can run
- Quadratic is a test: degree 2, set to 0, a ≠ 0.
- Name the given before the unknown
- The given is a, b, c. The unknown is x or the nature. Copy the signs before you hunt −2 and −3.
- One worked case is enough at this class
- One quadratic is enough. A page of the same 2-or-3 clone does not add a new D.
Practise Quadratic Equations
Reading is free and needs no account. Practice, mocks and progress live in the app.
- A 4-question practice set that ends the chapter
- 1 quick check with worked explanations
- Timed mocks scored with the real marking scheme
- Readiness tracked per topic, kept on your device