CBSE Class 10 Maths Quadratic Equations Guide for 2027: Roots, Discriminants and Word Problems

CBSE Class 10 Maths quadratic equations are easier to master when you know how to identify coefficients, choose a solving method, interpret the discriminant and translate word problems into algebra. This guide for the 2027 exam cycle explains the main techniques, common mistakes and a four-week revision plan. Use it with the prescribed textbook and verify the current CBSE syllabus and sample-paper guidance.

For each question, write the equation in standard form, identify the method, show the working and check the roots against the original equation. In word problems, also check whether each root is valid in the context.

What Is a Quadratic Equation?

A quadratic equation in one variable can be written as ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The highest power of the variable is two. If the coefficient of x² is zero, the equation is no longer quadratic, so check the value of a before applying a formula.

The standard form matters because it helps you identify the coefficients correctly. For example, in 2x² − 5x + 3 = 0, a = 2, b = −5 and c = 3. The sign belongs to the coefficient: b is negative five, not five. Rewriting an equation into standard form first prevents sign errors.

A quadratic equation may have two distinct real roots, one repeated real root or no real roots, depending on the discriminant. The roots are the values of x that make the equation true. You can solve many questions by factorisation, completing the square or the quadratic formula.

How to Identify a, b and c Correctly

Before solving, arrange all terms on one side so the other side is zero. Then compare the expression with ax² + bx + c = 0. Include signs and numerical coefficients exactly as written; if a term is missing, its coefficient is zero.

For 3x² + 7 = 4x, subtract 4x from both sides to get 3x² − 4x + 7 = 0. Therefore a = 3, b = −4 and c = 7. A frequent mistake is to use the original arrangement and overlook that the x-term changes sign when moved to the left.

Practise identifying coefficients without solving the equation. This small habit saves time in formula-based questions and word problems. If fractions appear, you may multiply the whole equation by a common denominator, but apply that operation to every term and preserve an equivalent equation.

Solving Quadratics by Factorisation

Factorisation rewrites a quadratic expression as a product of two linear factors. Once the equation is in the form (px + q)(rx + s) = 0, use the zero-product property: at least one factor must be zero. Solve each resulting linear equation and list both roots.

For x² − 5x + 6 = 0, find two numbers whose product is 6 and sum is −5: −2 and −3. Thus (x − 2)(x − 3) = 0, giving x = 2 or x = 3. Substitute either root into the original equation to check the result.

Factorisation is usually fastest when the coefficients are simple integers. If the factors are not obvious, do not waste too much time guessing; consider the discriminant or quadratic formula. In examination work, show the factorised form and both equations so the method is visible.

Splitting the Middle Term

For ax² + bx + c, the splitting-middle-term method looks for two terms whose coefficients add to b and whose product is ac. Split bx into those terms, group the four terms into pairs and factor each pair. The common binomial factor should emerge from both groups.

For 2x² + 7x + 3 = 0, the product ac is 6 and the sum b is 7. Split 7x as 6x + x: 2x² + 6x + x + 3. Grouping gives 2x(x + 3) + 1(x + 3), so (2x + 1)(x + 3) = 0.

If the chosen pair does not produce a common factor, check the product and sum again. Negative coefficients are especially prone to sign errors. This method is a form of factorisation, not a separate rule for finding roots, and the zero-product property is still the final step.

Quadratic Formula: A Reliable General Method

For ax² + bx + c = 0 with a ≠ 0, the roots are x = [−b ± √(b² − 4ac)]/(2a). The plus-minus sign represents two possible values. Substitute a, b and c with their signs in parentheses, calculate the discriminant, and simplify carefully.

The formula works even when simple integer factorisation is not obvious. Keep the denominator 2a outside the entire numerator. Do not accidentally divide only the square-root term or omit the negative sign before b. If the coefficients are fractions, simplifying the equation first may make the arithmetic easier.

A sound layout is: identify coefficients, write the formula, substitute, evaluate the discriminant, simplify each root and state the final answer. This sequence makes it easier to spot a sign mistake and shows the examiner how the result was obtained.

Discriminant and the Nature of Roots

The discriminant is D = b² − 4ac. If D is positive, the equation has two distinct real roots; if D is zero, it has equal real roots; and if D is negative, it has no real roots. These conclusions follow from whether the square root in the quadratic formula is positive, zero or not real.

Calculate the discriminant before solving fully when a question asks only for the nature of roots. For example, x² − 4x + 4 = 0 has D = 16 − 16 = 0, so its roots are equal. This shortcut saves work when only the classification is required.

If a parameter k appears in a, b or c, express D in terms of k and apply the requested condition. For equal roots, set D = 0; for distinct real roots, require D > 0; for no real roots, require D < 0. Also check any restrictions on the parameter given in the question.

Relationship Between Roots and Coefficients

If α and β are the roots of ax² + bx + c = 0, then α + β = −b/a and αβ = c/a. These relationships let you check a pair of roots or build an equation when the sum and product are known.

For x² − 7x + 10 = 0, the sum of roots should be 7 and the product should be 10. The roots 5 and 2 satisfy both relationships. If your calculated roots fail either check, revisit the algebra or the signs.

When constructing a monic quadratic with roots α and β, use x² − (α + β)x + αβ = 0. For a non-monic equation, multiply by a suitable non-zero constant if the question permits an equivalent equation. Avoid confusing the sum of roots with the negative of the coefficient b when a is not one.

Completing the Square

Completing the square transforms a quadratic expression into a perfect-square expression plus or minus a constant. For x² + bx + c = 0, move c to the other side and add (b/2)² to both sides so the left becomes (x + b/2)².

For x² + 6x + 5 = 0, write x² + 6x = −5, then add 9 to both sides: (x + 3)² = 4. Taking square roots gives x + 3 = ±2, so x = −1 or x = −5. Remember the positive and negative square roots.

This method is useful for understanding where the quadratic formula comes from and for equations that are easier to rearrange than factor. When the coefficient of x² is not one, divide through by that coefficient first, provided it is non-zero, and then complete the square.

Word Problems: Consecutive Integers

Read the problem carefully and define the unknown before forming an equation. If the smaller integer is n, the next consecutive integer is n + 1; for consecutive even or odd integers, the difference is two. Translate the stated sum or product into an equation rather than guessing numbers.

For example, if the product of two consecutive positive integers is 72, let them be n and n + 1. Then n(n + 1) = 72, or n² + n − 72 = 0. Factor to obtain candidate values and select the pair that satisfies the context, including the requirement that the integers are positive.

After solving, substitute the values into the original statement. Quadratic equations may produce a negative root even when the problem describes positive quantities. Do not discard a root until you explain which condition makes it unsuitable.

Word Problems: Area and Geometry

Geometry problems often connect the dimensions of a rectangle, a square or a path with an area condition. Define a dimension in terms of x, write the second dimension using the stated relationship, and use area = length × breadth. Expand and rearrange to form a quadratic equation.

For a rectangle, if the length is three metres more than the breadth x, write length as x + 3 and area as x(x + 3). If the given area is known, set x(x + 3) equal to that area and solve. Keep the physical units consistent throughout.

A mathematical root is not automatically a valid dimension. Lengths and widths must be positive, and the selected values must satisfy the original geometry condition. Draw a simple diagram and label the dimensions before writing the equation; this is particularly helpful when a border or path changes the outer dimensions.

Word Problems: Speed, Time and Distance

Speed problems can create quadratic equations when time changes with speed or when two journeys are compared. Use the relationship distance = speed × time, and remember that time = distance/speed when distance is fixed. Keep units consistent, such as kilometres with hours or metres with seconds.

If a person travels a fixed distance at speed v and later increases speed, the time decreases. A question may give the difference between two travel times; express both times as distance divided by their respective speeds, then use the condition to form an equation. Simplify carefully before multiplying through by denominators.

Check the meaning of the resulting roots. Speed and time are ordinarily positive in the context, so a negative candidate may not represent a physical solution. Substitute the chosen value into the original time relationship to ensure that it matches the stated time difference.

Word Problems: Numbers, Digits and Reversal

For a two-digit number with tens digit x and units digit y, the number is 10x + y. Its reversed form is 10y + x. Remember that digits are integers from 0 to 9, and the tens digit of a two-digit number cannot be zero.

If the digits satisfy a sum condition and the number is related to their product or reversed number, express each condition separately. A quadratic may appear after substituting one digit in terms of the other. Keep the digit constraints alongside the algebra so impossible values are not accepted.

Once you find candidate digits, check that each is an integer in the allowed range and that the original number has the required number of digits. Substitute into both the original and reversed expressions. This verification catches answers that satisfy one equation but not all conditions.

Building Equations from Real-Life Conditions

A word problem usually contains a relationship, a numerical condition and a quantity to find. Translate one sentence at a time. Assign a variable, express related quantities in terms of it, form the equation, bring all terms to one side and identify the quadratic coefficients.

Do not rush to expand every expression immediately. Keeping a factored form can make the relationship easier to understand, and expansion can be done once the equation is established. If a variable represents a measurement, include a note about its permitted range before solving.

After solving, test every candidate against the full story, not just the rearranged equation. Check whether the quantity must be positive, integral or within a given interval. A complete word-problem answer states the accepted value with its unit and briefly explains why any other root is rejected.

Common Algebra Errors and How to Catch Them

The most common mistakes are changing a sign incorrectly, forgetting to move all terms to one side, using the wrong coefficient in the discriminant and dropping the ± sign in the quadratic formula. Another error is cancelling terms across addition, which is not a valid algebraic operation.

Use a short check after each transformation: expand factors to confirm the original expression, recalculate b² − 4ac independently, and substitute each root into the original equation. If fractions are present, apply the same multiplication to every term and simplify only after preserving equality.

Keep your working aligned and write one transformation per line when the algebra is complex. In a timed test, neat working is not decoration; it helps you see sign errors and allows a marker to follow your method even if the final arithmetic is incorrect.

How to Check Roots Efficiently

A candidate root is correct if substituting it into ax² + bx + c produces zero. For integer roots, the sum and product relationships can provide a quick additional check. For approximate decimal roots, substitute and check that the result is close to zero, allowing for rounding.

If factorisation gives (x − 4)(x + 2) = 0, the candidates are 4 and −2. Expand the factors to ensure they reproduce the original quadratic. Then test the roots or use the coefficient relationships. This catches mistakes introduced when splitting the middle term.

For word problems, there is a second level of checking: confirm that the roots meet the story’s constraints. A negative length or non-integer digit is not a valid contextual answer even if it solves the algebraic equation. State the reason for selecting the final root.

Exam Strategy: Choosing the Fastest Method

Use factorisation when integer factors are clear, the quadratic formula when factorisation is awkward, and the discriminant when only the nature of roots is asked. Completing the square can be useful for derivations or when the expression naturally forms a perfect square. Choosing a suitable method saves time without compromising accuracy.

During practice, classify each question before solving: direct equation, parameter/discriminant, roots-and-coefficients relationship or word problem. That first decision helps you avoid lengthy algebra for a question that has a short route. Still show enough steps to justify the answer.

Keep time for verification. Recheck signs, both roots, the final unit and the contextual restrictions. If you are stuck trying to factorise, switch to a general method rather than repeatedly guessing. The goal is a correct, clearly presented solution, not loyalty to one technique.

A Four-Week Revision Plan for Quadratic Equations

In week one, revise standard form, coefficients, factorisation and splitting the middle term. Solve short sets with integer coefficients and verify every pair of roots. In week two, practise the quadratic formula, discriminant and nature of roots, including questions with an unknown parameter.

In week three, focus on sum and product of roots, completing the square and word problems involving integers, geometry and speed. Write the variable definition clearly for each story problem and check the physical or numerical constraints. Keep a list of errors you repeat.

In week four, attempt mixed timed sets and review mistakes by category: concept, sign, calculation, reading or method selection. Revisit weak question types after a short gap, then attempt a fresh set without notes. Check the current CBSE curriculum and sample-paper guidance before the exam.

NCERT and Official CBSE Resources

Use your prescribed NCERT Mathematics textbook as the base for definitions, examples and exercises. The official textbook portal is https://ncert.nic.in/textbook.php, and the CBSE Academic website https://cbseacademic.nic.in/ provides curriculum and sample-paper information. Confirm the relevant academic session rather than relying on an old syllabus or unofficial prediction.

Practice books and online question sets can add variety, but the core method and terminology should remain consistent with your course. When an answer key differs from your working, check the equation, factorisation and substitution step by step before assuming either result is correct.

The Infinity Home Tuitions shares learning options at https://theinfinityhometuitions.com/ and live classes at https://theinfinity.live. Students can compare available support according to subject needs and schedule. Bring attempted questions to a session so the focus stays on reasoning, error correction and independent practice.

Frequently Asked Questions

Which method is best for solving a quadratic equation?

Use factorisation when factors are clear, the quadratic formula for general cases and the discriminant when you only need to determine the nature of roots. Choose the method that is accurate and efficient for the given equation.

What does a zero discriminant mean?

When b² − 4ac equals zero, the quadratic has two equal real roots. A positive discriminant gives two distinct real roots, while a negative discriminant means there are no real roots.

How should I solve word problems?

Define the unknown, express related quantities in terms of it, form the equation, solve and check every root against the original context. Reject a root only when a stated condition makes it invalid.

Final Revision Checklist

• Write equations in standard form before identifying a, b and c.
• Practise factorisation and the quadratic formula.
• Know the discriminant conditions for the nature of roots.
• Use sum and product relationships to check roots.
• Translate word problems carefully and check contextual restrictions.
• Review mistakes and attempt timed mixed questions.