How to Form a Quadratic Equation from a Word Problem (Class 10)

In one line: to form a quadratic equation from a word problem, let x be the quantity you must find, write every other quantity in terms of x, translate the given condition into an equation, and simplify it into standard form ax² + bx + c = 0.

This skill — the NCERT calls it representing situations mathematically — is the first thing Chapter 4 asks of you, and it is tested directly (“represent the following situations in the form of quadratic equations”) before any solving begins. A quadratic equation appears whenever the condition in the problem multiplies two expressions that both contain x — an area (length × breadth), a product of two numbers, or a total cost (items × price each). That multiplication creates the x² term. If you want a refresher on what standard form means, see Standard Form of a Quadratic Equation first.

Here is the whole method in four steps: (1) Let x be what the question asks for → (2) Express the other quantities in terms of x → (3) Translate the condition into an equation → (4) Simplify to standard form. The worked examples below apply these same four steps to the three situation types the board exam loves most: area, product of numbers, and cost.

▶ Watch: Can you form the correct area equation? 🧮

The 4-step translation method

Use the same four steps for every problem, in the same order:

1. Let. Choose x = the quantity the question asks you to find (breadth, number, count of items). Write the “Let” statement explicitly — it earns a mark and anchors everything else.
2. Express. Write every other quantity in the problem in terms of x. “Length is one more than twice the breadth” becomes length = 2x + 1.
3. Translate. Turn the numerical condition into an equation: area = 480, product = 252, total cost = 216.
4. Simplify. Expand the brackets, bring all terms to one side, and write the result in standard form ax² + bx + c = 0 with descending powers.

Exam tip: In “represent mathematically” questions, the final line should always be the standard-form equation. Stopping at x(2x + 2) = 480 leaves the last mark on the table — finish the simplification.

Worked example 1 — an area problem

Problem: The area of a rectangular garden is 480 m². Its length (in metres) is two more than twice its breadth. Represent this situation as a quadratic equation.

• Let the breadth be x metres.
• Express: the length is (2x + 2) metres.
• Translate: area = length × breadth, so x(2x + 2) = 480.
• Simplify: 2x² + 2x = 480 → 2x² + 2x − 480 = 0 → dividing by 2, x² + x − 240 = 0.

The breadth of the garden satisfies the quadratic equation x² + x − 240 = 0. (Quick sense-check: a breadth of 15 m gives a length of 32 m and area 15 × 32 = 480 m² ✓ — so the equation is consistent with the story.)

Notice the last move: when every coefficient shares a common factor, divide it out. Smaller coefficients make the later factorisation step far friendlier.

Worked example 2 — a product-of-numbers problem

Problem: Meena and Ravi together have 40 stamps. Each of them gives away 4 stamps, and the product of the numbers of stamps they now have is 252. Represent this situation as a quadratic equation.

• Let Meena originally have x stamps.
• Express: Ravi has (40 − x) stamps. After giving away 4 each, Meena has (x − 4) and Ravi has (36 − x).
• Translate: (x − 4)(36 − x) = 252.
• Simplify: 36x − x² − 144 + 4x = 252 → −x² + 40x − 144 = 252 → −x² + 40x − 396 = 0 → multiplying by −1, x² − 40x + 396 = 0.

Meena’s original number of stamps satisfies x² − 40x + 396 = 0. Two habits shown here save marks everywhere: multiply the final equation by −1 so the x² term is positive, and expand (x − 4)(36 − x) term by term rather than by memory — sign slips inside such products are exactly what examfront’s Mistake Identification catches most often in this chapter.

Worked example 3 — a cost problem

Problem: A juice stall sells a certain number of cups in a day. The price of each cup (in ₹) is 30 minus the number of cups sold that day. On a day when the total collection was ₹216, represent the situation as a quadratic equation.

• Let the number of cups sold that day be x.
• Express: the price of each cup is ₹(30 − x).
• Translate: total collection = (cups) × (price each), so x(30 − x) = 216.
• Simplify: 30x − x² = 216 → −x² + 30x − 216 = 0 → x² − 30x + 216 = 0.

The number of cups sold satisfies x² − 30x + 216 = 0.

A quicker one to try yourself: the product of two consecutive positive integers is 182. Let the smaller be x; the next is x + 1; then x(x + 1) = 182 → x² + x − 182 = 0. If your equation matches, the 4-step method is settling in — a timed mixed set on examfront’s Topic Practice will make it automatic.

▶ Watch: Can you build the correct equation for consecutive even numbers? 🧮

Why these situations always produce x²

Each condition above multiplies two brackets that both contain x:

Contrast this with the word problems of Chapter 3, where conditions like “cost of 2 notebooks and 1 pen is ₹80” only add multiples of the variables — producing linear equations, not quadratics. The moment a problem multiplies two x-dependent quantities, expect degree 2.

Age problems (“the product of their ages 3 years from now…”) and speed–distance–time problems (“if the speed had been 8 km/h less…”) follow the same four steps but need an extra layer — shifted expressions like (x + 3) for future ages, or time written as distance/speed, which brings fractions into Step 3. We have deliberately kept those heavier variations out of this guide: the full set of age, speed and two-condition problems, with step-by-step personalised help where your translation goes wrong, lives inside examfront, and the Practice Companion is built for exactly that jump in difficulty.

▶ Watch: Can you turn the reciprocal relation into a quadratic? 🧮

Key Takeaways

• Forming a quadratic equation from a word problem is a 4-step translation: Let → Express → Translate → Simplify to standard form.
• Always let x = the quantity the question asks for, and write every other quantity in terms of x.
• Area, product-of-numbers and cost conditions all multiply two x-dependent expressions, which is what creates the x² term.
• Finish every answer in standard form ax² + bx + c = 0 — divide out common factors and make the x² coefficient positive.
• “Represent mathematically” questions award full marks for the correct equation; solving comes later in the chapter.

Quick Facts

• Method: Let → Express → Translate → Simplify (4 steps).
• Choose x as: the quantity to be found.
• x² appears when: the condition multiplies two expressions containing x.
• Final answer format: standard form ax² + bx + c = 0, descending powers, positive x² coefficient preferred.
• Typical situation types: area of a rectangle, product of two numbers, total cost/collection.
• Chapter: Quadratic Equations · Class: 10 · Subject: Maths · Board: CBSE.

Common Mistakes

1. Letting x be the wrong quantity. Choosing x = length when the question asks for breadth doubles the algebra and invites errors. Let x be what is asked; express the rest in terms of it.
2. Translating the comparison backwards. “Length is two more than twice the breadth” is 2x + 2, not 2(x + 2). Translate word by word: twice the breadth (2x), two more (+ 2).
3. Sign slips while expanding brackets like (x − 4)(36 − x). The −x × −4 and cross terms are where marks vanish. Expand term by term and combine carefully.
4. Stopping before standard form. Leaving the answer as x(30 − x) = 216 is incomplete. Expand, bring everything to one side, and write ax² + bx + c = 0.
5. Forgetting units and the “Let” statement. “Let the breadth be x metres” earns presentation marks and prevents unit confusion; omitting it makes the solution harder to follow.

FAQ

Q. How do you form a quadratic equation from a word problem? Four steps: let x be the quantity to find; express every other quantity in terms of x; translate the given condition (area, product, total cost) into an equation; then expand and simplify into standard form ax² + bx + c = 0.

Q. Which quantity should I take as x? The one the question asks for — the breadth, the number, the count of items. Then build the other quantities from it (length = 2x + 2, the other number = 40 − x). One well-chosen variable keeps the whole translation direct.

Q. Why do area and product problems give quadratic equations? Because the condition multiplies two expressions that each contain x — length × breadth or number × number — creating an x² term, so the equation has degree 2.

Q. Do I have to solve the equation after forming it? Only if asked. “Represent the situation mathematically” questions give full marks for the correct standard-form equation. Solving by factorisation is the next section of the chapter — see the chapter guide for the roadmap.

Q. How do I check my equation is correct? Re-read the problem and verify each expression matches its phrase, then re-derive the condition line and redo the expansion once. A quick consistency check (do sensible numbers fit?) catches most translation errors.

Related Concepts

• Standard Form of a Quadratic Equation (Class 10): the definition and the a ≠ 0 condition your final line must satisfy.
• Quadratic Equations — Chapter Guide: where forming fits before factorisation and the quadratic formula.
• Linear-Equations Word Problems (Class 10): the degree-1 cousin — compare which conditions stay linear.
• Polynomials — Chapter Guide: the quadratic polynomial behind every quadratic equation.

Can you translate any story into ax² + bx + c = 0 without hesitation? Prove it with a Chapter Quiz on examfront — and let Strength & Weakness Analysis show whether translation or expansion is costing you marks. Start on examfront →