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Quadratics in context

Quadratic word problems — factor, then pick the root they asked for

Quadratic word problems describe a product or a sum of squares and leave you to build the equation. Translate the sentence into standard form, factor it into two brackets, and read both roots. The last step decides the mark: questions ask for the larger root, the smaller integer, or the sum, and all of those values appear on the option list.

Most of these factor cleanly, so hunting for the quadratic formula usually signals a translation error rather than a hard equation. Two consecutive integers with a product of 72 give n² + n − 72 = 0, which factors as (n + 9)(n − 8). The pair is 8 and 9, and the question then asks for one specific thing about them.

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How to solve quadratic word problems in four steps

  1. Express every quantity from one unknown

    Two consecutive integers are n and n + 1; two numbers differing by 3 are n and n + 3. One letter keeps the translation to a single equation.

  2. Move everything to one side

    A product of 72 becomes n(n + 1) = 72, then n² + n − 72 = 0. Standard form is what makes the factor pair visible.

  3. Find the factor pair that multiplies and adds correctly

    For n² + n − 72 you need two numbers multiplying to −72 and adding to 1: that is 9 and −8, giving (n + 9)(n − 8) = 0.

  4. Choose the root the question asked for

    Roots of −9 and 8 both satisfy the equation. If the question says positive integers, discard the negative, then check whether it wants the larger, the smaller or the sum.

Translating the sentence into a quadratic

The sentenceThe equation
Two consecutive integers, product 72n² + n − 72 = 0
Two consecutive even integers, product 48n² + 2n − 48 = 0
A number plus its square is 30n² + n − 30 = 0
Two numbers differ by 3, product 40n² + 3n − 40 = 0

3 sample quadratic word problems, with worked solutions

Generated by the same engine that mints the Quant Daily daily. Each answer is computed from the numbers printed in the question, and the walkthrough below each one is the engine's own working — not a solution written afterwards. Reload this page's live drill and you get different numbers.

Question 1

The product of two consecutive positive integers is 182.

What is the smaller integer?

  1. -13
  2. 13
  3. 27
  4. 14
  5. -14
Show the worked solution
  1. n(n+1) = 182.
  2. The integers are 13 and 14.
  3. Answer: 13.

Question 2

A number squared minus 21 times the number plus 104 equals 0.

What is the larger solution?

  1. 12
  2. 11
  3. 8
  4. 23
  5. 13
Show the worked solution
  1. x² − 21x + 104 = 0.
  2. (x − 8)(x − 13) = 0.
  3. Roots 8 and 13. Larger: 13.
  4. Answer: 13.

Question 3

Two numbers have product 117 and one is 9.

What is the sum of the two numbers?

  1. 13
  2. 9
  3. 22
  4. 23
  5. 117
Show the worked solution
  1. The other number is 117 ÷ 9 = 13.
  2. Sum = 9 + 13 = 22.
  3. Answer: 22.

Common mistakes this drill is built from

The wrong options are not random numbers. Each one is the result of a specific careless error for this topic, so picking one tells you which habit is costing you marks.

Selecting the discarded root

A quadratic returns two roots and the context usually rules one out. A negative root is not an answer to a question about positive integers.

Answering with the wrong member of the pair

Once you have 8 and 9, the question decides which one counts. Both appear as options, as does their sum.

Reaching for the formula on a factorable quadratic

These are built to factor. If no integer pair works, re-read the sentence before grinding through the discriminant.

Why your answer came out wrong

The specific wrong numbers this topic produces, and what each one tells you about the step you took. If you have just got a question wrong and want to know which habit did it, start here.

Two consecutive integers have a product of 72 — is the answer 8 and 9 or −9 and −8?

Both pairs genuinely multiply to 72, so the wording decides. If the question specifies positive integers, the answer is 8 and 9. If it does not, the negative pair is equally valid, which is why these questions almost always say 'positive'.

If the roots are 8 and 9, is the answer 17?

Only if the question asked for their sum. If it asked for the larger integer, the answer is 9, and if it asked for the smaller, it is 8. All three values are typically on the option list, so the deciding step is re-reading what was requested.

FAQ

How do I set up two consecutive integers with a given product?

Call them n and n + 1, so their product is n² + n. Setting that equal to 72 and moving everything across gives n² + n − 72 = 0, which factors as (n + 9)(n − 8). The positive solution is 8, so the pair is 8 and 9.

Why does my quadratic give two answers when the question wants one?

A quadratic has two roots by nature, and the wording of the problem eliminates one. Lengths, counts and positive integers rule out negatives, while a question about the larger of two values picks between two valid roots. Read the final sentence before selecting.

Do I need the quadratic formula for these?

Rarely. Exam-style word problems are built around clean factor pairs, so scanning for two numbers that multiply to the constant and add to the middle coefficient is faster and less error-prone. If nothing factors, the translation is usually where the mistake is.

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