CDS Mathematics · Quadratic Equations

Common Roots

Two quadratics share a root when one value of x satisfies both; subtracting the equations removes the x² term and finds it.

All Quadratic Equations notes CDS Mathematics strategy

Why this matters

Six PYQs, all short. Either one of the two equations factorises and its roots can be tried in the other, or neither does and subtracting the equations gives the common root directly.

Concept 1 of 2: Subtract to find the common root

At the common root both equations hold, so their difference also holds. Both have

x^2

with coefficient

1

, so the difference is linear and gives the root at once.

Definition

If $x^2 + px + q = 0$ and $x^2 + qx + p = 0$ share a root and $p \ne q$ : subtracting gives $(p - q)(x - 1) = 0$ , so the root is $1$ and $1 + p + q = 0$ .
A common factor $(x - r)$ means a common root $r$ : substitute $r$ into both.
If the common root is given, substitute it into each equation separately.

Common root by subtraction

(x^2 + px + q) - (x^2 + qx + p) = (p - q)(x - 1)

Worked example

x^2 - 3x + k = 0

and

x^2 - 5x + 2k = 0

have a common root,

k \ne 0

. Find

k

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2025 · CDS (I) 2025 — Elementary Mathematics · Q80Moderate

Example 1 · Quadratic Equations · Common Roots

The equations

x^2 + px + q = 0

and

x^2 + qx + p = 0

(p \ne q)

have a common root. What is the value of

(p + q)

Discard the value the stem rules out

The subtraction often gives two candidates, one of which makes a parameter zero or makes the two equations identical. If the stem says

k \ne 0

p \ne q

, that candidate is gone.

Drill 3 more on subtract to find the common root

Concept 2 of 2: Factor one and try its roots

If one of the two equations factorises, the common root must be one of its two roots. Try each in the other equation; each choice gives one value of the unknown.

Definition

Factorise the equation with numbers only.
Substitute each of its roots into the other equation and solve for the unknown.
Both roots may work, giving two answers; the options often list them as a pair.
In data sufficiency, a common root needs BOTH equations, so neither alone can decide it.

Common root from a factorised equation

(x - r_1)(x - r_2) = 0 \;\Rightarrow\; \text{common root is } r_1 \text{ or } r_2

Worked example

x^2 - 3x + 2 = 0

and

x^2 + kx + 4 = 0

have a common root. Find

k

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (I) 2019 — Elementary Mathematics · Q19Moderate

Example 2 · Quadratic Equations · Common Roots

If the equations

x^2 + 5x + 6 = 0

and

x^2 + kx + 1 = 0

have a common root, then what is the value of

k

Two answers, not one

Each root of the factorised equation can be the shared one, so there are usually two values of the unknown. An option giving only one of them is incomplete.

Drill 1 more on factor one and try its roots

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)

Subtract to find the common root
Common root by subtraction
$(x^2 + px + q) - (x^2 + qx + p) = (p - q)(x - 1)$
Factor one and try its roots
Common root from a factorised equation
$(x - r_1)(x - r_2) = 0 \;\Rightarrow\; \text{common root is } r_1 \text{ or } r_2$

Watch out for (2)

Discard the value the stem rules out
→ Subtract to find the common root
Two answers, not one
→ Factor one and try its roots

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