Class 10 Maths · Original EZT practice
Quadratic Equations (Class 10 Chapter 4): summary and practice
Quadratic Equations takes the polynomials of Chapter 2 and solves them: the standard form ax^2 + bx + c = 0, splitting the middle term to factorise, the quadratic formula, and the discriminant b^2 - 4ac that tells you the nature of the roots before you solve. All questions are original EZT tutor practice following the chapter's own section order.
6 original worked examples, 6 fresh self-check questions and 4 mixed questions, with explained answers. These are not copied textbook exercises, an official chapter index or a full answer key. No affiliation with NCERT is claimed.
Official sources for topic and edition context, checked 21 October 2026:
1. Introduction
Chapter 2 studied polynomials; now they become equations. Area problems, speed problems and product-of-consecutive-number problems all turn into a single quadratic equation once you name the unknown.
2. Quadratic equations and standard form
A quadratic equation in x is ax^2 + bx + c = 0 with a not zero - otherwise the x^2 term vanishes and the equation is linear. Before reading off a, b and c, move every term to one side so the other side is zero.
3. Solution by factorisation
Split the middle term bx into two parts whose product is a x c x x^2, factor by grouping, and set each factor to zero. A quadratic has at most two real roots, and a value is a root only if it makes the left side exactly zero.
4. Quadratic formula and nature of roots
Section 4.4 gives the formula x = (-b +/- sqrt(b^2 - 4ac)) / 2a. The discriminant D = b^2 - 4ac decides everything: D > 0 two distinct real roots, D = 0 two equal real roots, D < 0 no real roots.
Choose, solve, check, then try a fresh question.
Before calculating, read the method prompt and name the condition that makes it useful. Write your own attempt. Open the steps only when you need them, and use the check to catch a wrong sign, unit or interpretation. Finish with the fresh question before reading its answer.
Solve by splitting the middle term
Choose the method: Find two numbers with sum b and product a x c, split, group, factor.
EZT practice question
Solve x^2 - 5x + 6 = 0.
Show the worked solution
- Need two numbers adding to -5 with product 6: -2 and -3.
- x^2 - 2x - 3x + 6 = x(x - 2) - 3(x - 2).
- (x - 2)(x - 3) = 0.
Answer: x = 2 or x = 3.
Check: Substitute back: 4 - 10 + 6 = 0 and 9 - 15 + 6 = 0 - both make the left side zero, so both are roots.
Watch for this: The product must be a x c (here 1 x 6) - splitting with the wrong pair never factorises.
Now try it yourself
Solve x^2 - 7x + 12 = 0.
Check your answer
x = 3 or x = 4.
-3 and -4 add to -7 and multiply to 12: (x - 3)(x - 4) = 0.
Read the discriminant first
Choose the method: D = b^2 - 4ac; D > 0 means two distinct real roots.
EZT practice question
For x^2 - 5x + 6 = 0, compute the discriminant and state the nature of the roots.
Show the worked solution
- a = 1, b = -5, c = 6.
- D = (-5)^2 - 4 x 1 x 6 = 25 - 24.
- = 1 > 0.
Answer: D = 1, so two distinct real roots (which are 2 and 3).
Check: D is a perfect square, so the roots come out rational - the factorisation in the previous example confirms it.
Watch for this: Square b WITH its sign: (-5)^2 is 25, not -25.
Now try it yourself
Discriminant of x^2 - 7x + 12 = 0?
Check your answer
D = 1, two distinct real roots.
49 - 48 = 1.
Equal roots when D = 0
Choose the method: D = 0 means two equal real roots - one repeated value.
EZT practice question
Show that 2x^2 - 4x + 2 = 0 has equal roots, and find them.
Show the worked solution
- a = 2, b = -4, c = 2.
- D = 16 - 4 x 2 x 2 = 16 - 16 = 0.
- The equation is 2(x - 1)^2 = 0.
Answer: D = 0, equal roots: x = 1, 1.
Check: x = (-b)/2a = 4/4 = 1 matches the factorisation - when D = 0 the formula collapses to one value.
Watch for this: Say equal roots, not one root - a quadratic always carries two roots counting repetition.
Now try it yourself
Discriminant of x^2 + 6x + 9 = 0?
Check your answer
D = 0, x = -3, -3.
36 - 36 = 0 and the equation is (x + 3)^2 = 0.
No real roots when D < 0
Choose the method: D < 0 means no real roots - the graph never crosses the x-axis.
EZT practice question
Does x^2 + x + 1 = 0 have real roots?
Show the worked solution
- a = 1, b = 1, c = 1.
- D = 1 - 4 x 1 x 1 = 1 - 4.
- = -3 < 0.
Answer: D = -3, so the equation has no real roots.
Check: x^2 + x + 1 = (x + 1/2)^2 + 3/4, always at least 3/4 - it can never reach zero, which matches.
Watch for this: Say no REAL roots, not no solution - complex roots exist, they are just beyond this chapter.
Now try it yourself
Does x^2 + 2x + 5 = 0 have real roots?
Check your answer
No - D = -16.
4 - 20 = -16 < 0.
Set up the standard form before solving
Choose the method: Bring every term to one side, then read a, b, c.
EZT practice question
Write (x + 2)(x - 3) = 4 in standard form and identify a, b, c.
Show the worked solution
- Expand: x^2 - x - 6 = 4.
- Move 4 across: x^2 - x - 10 = 0.
- Read off the coefficients.
Answer: a = 1, b = -1, c = -10.
Check: Only now does D make sense: 1 + 40 = 41 > 0, two distinct real roots.
Watch for this: Reading c = -6 before moving the 4 across gives the wrong discriminant entirely.
Now try it yourself
Standard form of x(x + 3) = 10?
Check your answer
x^2 + 3x - 10 = 0.
Expand, then subtract 10 from both sides.
Apply the quadratic formula
Choose the method: x = (-b +/- sqrt D) / 2a once D is known.
EZT practice question
Solve x^2 + 7x - 60 = 0 using the quadratic formula (as the chapter does inside 4.4).
Show the worked solution
- a = 1, b = 7, c = -60.
- D = 49 + 240 = 289, and sqrt 289 = 17.
- x = (-7 +/- 17) / 2.
Answer: x = 5 or x = -12.
Check: (x - 5)(x + 12) = x^2 + 7x - 60 - expanding returns the original equation.
Watch for this: Watch the sign of c: -4ac is -4 x 1 x (-60) = +240, not -240.
Now try it yourself
Solve x^2 - 3x - 10 = 0 by the formula.
Check your answer
x = 5 or x = -2.
D = 9 + 40 = 49, so x = (3 +/- 7) / 2.
Try a mixed set
Choose a method or explanation before opening each answer.
1. Standard form?
Show answer and reason
ax^2 + bx + c = 0, a not 0.
Without x^2 it is linear.
2. Discriminant?
Show answer and reason
D = b^2 - 4ac.
It sits under the root in the formula.
3. D > 0, = 0, < 0?
Show answer and reason
Two roots, equal roots, no real roots.
The root count follows the sign.
4. Formula?
Show answer and reason
x = (-b +/- sqrt D) / 2a.
Complete the square once, use forever.
Try it yourself
Work these on paper before checking any notes. Answers are deliberately not shown here.
- Solve x^2 - 7x + 12 = 0 by factorisation.
- Compute D for x^2 + 6x + 9 = 0 and name the roots.
- Does x^2 + 2x + 5 = 0 have real roots? Say why in one line.
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Questions about this practice page
- What does the discriminant tell you without solving?
- Its sign alone predicts the roots: D > 0 gives two distinct real roots, D = 0 gives two equal real roots, and D < 0 means the equation has no real roots at all. Computing D first saves you from chasing roots that do not exist.
- How do I split the middle term reliably?
- Look for two numbers that ADD to b and MULTIPLY to a x c. Rewrite bx as their sum, group the four terms in pairs, and factor each pair - the same bracket should appear twice, giving (x - p)(x - q) = 0.
- Is the quadratic formula in this chapter?
- Yes - in the Reprint 2026-27 the formula x = (-b +/- sqrt(b^2 - 4ac)) / 2a appears inside section 4.4, Nature of Roots, where the chapter uses it to read the root types straight from the discriminant.
- Why must one side be zero before reading a, b, c?
- Because the coefficients are defined only for ax^2 + bx + c = 0. A stray term on the right changes c - and with it the discriminant, the roots, and every step after.
- Are these the NCERT exercise questions for Quadratic Equations (Class 10)?
- No. EZT pages are original practice following the chapter's section order - use them to prepare for the textbook exercises, not as a substitute for your book.