CDS Mathematics · Quadratic Equations

Roots in a Given Relation

When the roots are tied together — in a ratio, reciprocal, or equal to the coefficients themselves — write them in that form and apply the sum and product.

All Quadratic Equations notes CDS Mathematics strategy

Why this matters

Ten PYQs. Every one starts the same way: name the roots so that the relation is built in (kt and t, t and 1/t, or p and q themselves), then write the sum and product and eliminate. The 'roots are p and q' question has appeared in four papers.

Concept 1 of 2: Roots in a ratio

If the roots are in the ratio

m : n

, call them

mt

and

nt

. The sum gives

t

, the product gives

t^2

, and squaring the first to match the second removes

t

Definition

For $ax^2 + bx + c = 0$ :

roots in the ratio $m : n$ $\iff$ $mn\,b^2 = (m + n)^2\,ac$ ;
one root twice the other ( $m : n = 2 : 1$ ): $2b^2 = 9ac$ ;
reciprocal roots (product $1$ ): $c = a$ ;
equal in size, opposite in sign (sum $0$ ): $b = 0$ .

Roots in the ratio m : n

mn\,b^2 = (m + n)^2\,ac

Worked example

One root of

x^2 - kx + 18 = 0

is twice the other. Find the positive value of

k

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2017 · CDS (I) 2017 — Elementary Mathematics · Q7Moderate

Example 1 · Quadratic Equations · Roots in a Given Relation

If one root of

(a^2 - 5a + 3) x^2 + (3a - 1) x + 2 = 0

is twice the other, then what is the value of 'a' ?

Keep both signs of t

t^2

from the product gives two values of

t

, and each gives a different coefficient. The question usually asks for 'the positive value'; make sure you pick the

t

that produces it.

Drill 3 more on roots in a ratio

Concept 2 of 2: When the roots are the coefficients

p

and

q

are both the coefficients and the roots of

x^2 + px + q = 0

, the sum and product give two equations in

p

and

q

. The product equation usually factorises, which splits the problem into cases.

Definition

For $x^2 + px + q = 0$ with roots $p$ and $q$ :

sum: $p + q = -p$ , so $q = -2p$ ;
product: $pq = q$ , so $q(p - 1) = 0$ ;
hence $p = q = 0$ , or $p = 1$ , $q = -2$ . A condition like $q \ne 0$ picks the second.

Shifted roots: if the roots of one equation increased by $k$ are the roots of another, compare sums ( $2k$ apart) and products.

Roots p and q of x² + px + q = 0

p + q = -p, \quad pq = q \;\Rightarrow\; (p, q) = (0, 0) \text{ or } (1, -2)

Worked example

The roots of

x^2 + ax + b = 0

are

2a

and

b

, with

b \ne 0

. Find

a

and

b

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (I) 2019 — Elementary Mathematics · Q89Moderate

Example 2 · Quadratic Equations · Roots in a Given Relation

The equation

x^2 + px + q = 0

has roots equal to

p

and

q

where

q \ne 0

. What are the values of

p

and

q

respectively ?

Do not divide by a letter that can be zero

From

pq = q

you may conclude

p = 1

only when

q \ne 0

. If the question does not say so,

p = q = 0

is also a solution, and an option like '

p = 0

1

' becomes the correct one.

Drill 5 more on when the roots are the coefficients

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)

Roots in a ratio
Roots in the ratio m : n
$mn\,b^2 = (m + n)^2\,ac$
When the roots are the coefficients
Roots p and q of x² + px + q = 0
$p + q = -p, \quad pq = q \;\Rightarrow\; (p, q) = (0, 0) \text{ or } (1, -2)$

Watch out for (2)

Keep both signs of t
→ Roots in a ratio
Do not divide by a letter that can be zero
→ When the roots are the coefficients

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 10 questions