CDS Mathematics · Quadratic Equations

Solving and Forming Quadratic Equations

A quadratic is fixed by the sum and the product of its roots, and when its coefficients add to zero one root is 1.

All Quadratic Equations notes CDS Mathematics strategy

Why this matters

Fifteen PYQs. Two moves cover almost all of them: write the equation as x² − (sum)x + (product) = 0, and check whether the coefficients add to zero before doing any algebra. Irrational roots come in conjugate pairs, which answers the 'other root' questions at once.

Concept 1 of 2: Forming an equation from its roots

Expanding

(x - \alpha)(x - \beta)

gives

x^2 - (\alpha + \beta)x + \alpha\beta

. So the sum and the product are all you need to write the equation, and a change to the roots is a change to that sum and product.

Definition

The equation with roots $\alpha, \beta$ is $x^2 - (\alpha + \beta)x + \alpha\beta = 0$ , or any non-zero multiple of it.
For $ax^2 + bx + c = 0$ : $\alpha + \beta = -\dfrac ba$ and $\alpha\beta = \dfrac ca$ .
Copying errors: a wrong constant term leaves the SUM right; a wrong $x$ -coefficient leaves the PRODUCT right.
With rational coefficients, an irrational root $p + \sqrt q$ comes with its conjugate $p - \sqrt q$ .
To find an equation satisfied by a new quantity $y$ , express $x$ through $y$ and substitute.

Equation from sum and product

x^2 - (\alpha + \beta)x + \alpha\beta = 0

Worked example

The roots of an equation have sum

5

and product

6

. Form the equation whose roots are each one more.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2017 · CDS (I) 2017 — Elementary Mathematics · Q45Moderate

Example 1 · Quadratic Equations · Solving and Forming Quadratic Equations

Aman and Alok attempted to solve a quadratic equation. Aman made a mistake in writing down the constant term and ended up in roots

(4, 3)

. Alok made a mistake in writing down the coefficient of

x

to get roots

(3, 2)

. The correct roots of the equation are

The sign of the sum

The

x

-coefficient is MINUS the sum. Roots with sum

2

give

x^2 - 2x + \ldots

, not

x^2 + 2x + \ldots

, and the wrong-sign version is always among the options.

Drill 7 more on forming an equation from its roots

Concept 2 of 2: When the coefficients add to zero

Putting

x = 1

into

ax^2 + bx + c

gives

a + b + c

. If that is zero,

1

is a root, and the other root is the product

\dfrac ca

. Letter-heavy equations are built this way on purpose.

Definition

If $a + b + c = 0$ , the roots of $ax^2 + bx + c = 0$ are $1$ and $\dfrac ca$ .
If $a - b + c = 0$ , the roots are $-1$ and $-\dfrac ca$ .
If a root is given, substitute it to find an unknown coefficient, then use the sum or product for the other root.
Otherwise factorise, or use $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .

Root 1

a + b + c = 0 \;\Rightarrow\; x = 1,\ \dfrac ca

Worked example

Find the roots of

(p - q)x^2 + (q - r)x + (r - p) = 0

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2025 · CDS (II) 2025 — Elementary Mathematics · Q25Moderate

Example 2 · Quadratic Equations · Solving and Forming Quadratic Equations

\left(\frac{a-b}{2}\right)x^2-\left(\frac{a+b}{2}\right)x+b=0

, then what are the roots of this equation?

Clear the fractions first

The coefficient-sum test works on the equation as it stands, fractions and all, but the product

\dfrac ca

is easier to read after multiplying through. Do not multiply only some of the terms.

Drill 6 more on when the coefficients add to zero

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)

Forming an equation from its roots
Equation from sum and product
$x^2 - (\alpha + \beta)x + \alpha\beta = 0$
When the coefficients add to zero
Root 1
$a + b + c = 0 \;\Rightarrow\; x = 1,\ \dfrac ca$

Watch out for (2)

The sign of the sum
→ Forming an equation from its roots
Clear the fractions first
→ When the coefficients add to zero

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 15 questions