CDS Mathematics · Quadratic Equations

Word Problems with Quadratics

Name the unknown, turn each sentence into an equation, solve the quadratic, and keep only the root the situation allows.

All Quadratic Equations notes CDS Mathematics strategy

Why this matters

Six PYQs, mostly MODERATE. The equation is always short; marks are lost in the last step, by keeping a negative length or a non-natural number, or by answering the wrong quantity (the sum of the numbers, not the smallest one).

Concept 1 of 2: Number problems

Consecutive numbers are

n, n + 1, n + 2

; consecutive even or odd numbers, and 'alternate' numbers, are

n, n + 2, n + 4

. With the right naming, the condition becomes a quadratic in

n

Definition

Consecutive: $n, n + 1, \ldots$ . Alternate or consecutive even/odd: $n, n + 2, \ldots$ .
A number and its square: $n + n^2$ ; a number and its reciprocal: $n + \dfrac1n$ .
Keep the root the question allows (natural, positive, integer). Both roots may be valid if nothing rules one out.
Answer the quantity asked for — often the sum or difference, not $n$ itself.

Sum of squares of consecutive numbers

n^2 + (n + 1)^2 = 2n^2 + 2n + 1

Worked example

The sum of the squares of three consecutive natural numbers is

149

. Find their sum.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (I) 2019 — Elementary Mathematics · Q88Moderate

Example 1 · Quadratic Equations · Word Problems and Applications

The sum of the squares of four consecutive natural numbers is 294. What is the sum of the numbers ?

Both roots can be right

'The sum of a number and its square is

30

' has two answers,

5

and

-6

, because nothing says the number is positive. Reject a root only when the stem rules it out.

Drill 3 more on number problems

Concept 2 of 2: Lengths and prices

A fixed total split in two ways gives two fractions whose difference is known. Clearing the denominators leaves a quadratic in the unknown length or quantity.

Definition

Price per unit $= \dfrac{\text{total}}{\text{quantity}}$ . 'Buy $2$ more for the same money and pay $r$ less per unit': $\dfrac Tn - \dfrac{T}{n + 2} = r$ , so $\dfrac{2T}{n(n + 2)} = r$ .
A point dividing a segment: if $AC = x$ then $CB = \text{length} - x$ ; keep the root that lies inside the segment.
Lengths, times and quantities are positive; drop a negative root.

Same total, two prices

\dfrac{T}{n} - \dfrac{T}{n + k} = r \;\Rightarrow\; n(n + k) = \dfrac{kT}{r}

Worked example

Some pens cost Rs. 180. With

3

more pens for the same money, each would cost Rs. 3 less. How many pens were bought?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2022 · CDS (I) 2022 — Elementary Mathematics · Q37Moderate

Example 2 · Quadratic Equations · Word Problems and Applications

A piece of cloth costs Rs. 10,000. If a 2 m longer piece of the same cloth is purchased for the same amount, it would cost Rs. 250 less per metre. What is the original length of the piece of cloth?

Keep the root inside the segment

When a point

C

divides a segment of length

4

, a root like

6 + 2\sqrt5

is longer than the segment itself. Only the root between

0

and the length is a real position.

Drill 1 more on lengths and prices

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Number problems
Sum of squares of consecutive numbers
$n^2 + (n + 1)^2 = 2n^2 + 2n + 1$
Lengths and prices
Same total, two prices
$\dfrac{T}{n} - \dfrac{T}{n + k} = r \;\Rightarrow\; n(n + k) = \dfrac{kT}{r}$

Watch out for (2)

Both roots can be right
→ Number problems
Keep the root inside the segment
→ Lengths and prices

Test yourself on Quadratic Equations

20 past CDS questions from this chapter, timed at 24 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Quadratic Equations chapter test Drill all 6 questions