Worked example 1
Question: Find the zeros of 2x²−7x+3 and verify the relationships.
Solution: Factor: 2x²−7x+3=(2x−1)(x−3). Zeros are 1/2 and 3. Sum=7/2=−(−7)/2; product=3/2=c/a.
What this chapter covers
Connect polynomial zeros with the x-intercepts of its graph and verify the relationship between zeros and coefficients of a quadratic polynomial.
Zeros and graphs
A zero α of p(x) satisfies p(α)=0. On the graph y=p(x), zeros are the x-coordinates where the curve meets or touches the x-axis. A quadratic graph may have two, one or no real x-intercepts.
Zeros and coefficients
For ax²+bx+c with zeros α and β: α+β = −b/a and αβ = c/a. These relationships can check a factorisation or help form a quadratic polynomial from its zeros.
Key formulae: Quadratic p(x)=ax²+bx+c: sum of zeros = −b/a; product of zeros = c/a. Monic polynomial with zeros α,β: x²−(α+β)x+αβ.
Worked examples: CBSE question patterns
These are original practice questions modeled on common board-exam patterns; they are not reproduced past-paper questions.
Worked example 2
Question: Form a quadratic polynomial whose zeros are 4 and −2.
Solution: Sum=2 and product=−8. A monic polynomial is x²−2x−8. Its factorisation (x−4)(x+2) confirms the zeros.
Practical activity / investigation
Plot y=x²−4 using a table for x=−3 to 3 on graph paper. Mark where the curve crosses the x-axis. Then factor the polynomial and compare the algebraic zeros with the graph.
Record: Write the aim, materials, steps, observations and conclusion in your notebook. Follow teacher supervision for any classroom activity.
Quick practice with answers
Try each question before opening its solution.
Practice 1: Find the zeros of x²−9.
x²−9=(x−3)(x+3), so the zeros are 3 and −3.
Practice 2: For 3x²+5x−2, state the sum and product of zeros.
Sum=−5/3; product=−2/3.
Practice 3: Write a monic quadratic with zeros −1 and 6.
Sum=5, product=−6; polynomial x²−5x−6.
Revision checklist
- Can I explain the key idea without memorising a worked solution?
- Have I selected the right formula and written units where needed?
- Can I show each algebraic or geometric step clearly?
- Can I check my answer by substitution, estimation or a diagram?
Aligned to the Class X Mathematics curriculum for 2026–27 and the NCERT Mathematics textbook. Always confirm any school-specific internal assessment with your teacher.