Quadratic Equations

A ball thrown up rises, pauses and falls — its height follows a quadratic equation. If a square garden’s area is given and the side is asked, you get $x^{2} = number$ . Quadratic means the highest power of the unknown is 2. Such an equation has two roots (solutions) — sometimes two different, sometimes equal, and sometimes none among real numbers. One formula $\frac{- b \pm b ^{2} - 4 a c}{2 a}$ solves them all, but factorisation is often faster.

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}, α + β = - \frac{b}{a}, α β = \frac{c}{a}

The three main formulas

1

The standard form

a x^{2} + b x + c = 0

,

a \neq = 0

Spot whether an equation is quadratic and read $a, b, c$ in standard form.

Recognising a quadratic

Simplify first

An equation that looks quadratic can become linear on simplifying, and vice versa.

Standard form: $a x^{2} + b x + c = 0$ ; $a$ cannot be zero (it would become linear).
Open brackets and collect terms — if $x^{2}$ cancels the equation stays linear.
Root: a value that satisfies the equation. A quadratic has exactly two roots (possibly equal).
If $α$ is a root then $(x - α)$ is a factor.

Example 1 · Basic

Is

(x - 2) (x + 3) = x^{2} - 5

quadratic or linear?

Solution, step by step

1The left side is $x^{2} + x - 6$ ; the $x^{2}$ cancels with the right side.
2What is left is $x - 6 = - 5$ ⟹ $x = 1$ : it is linear

Answer Linear

BasicTry it yourself first

Which of these is a quadratic equation?

StandardTry it yourself first

Which of these is a quadratic equation?

StandardTry it yourself first

Which of these is a quadratic equation?

StandardTry it yourself first

Which of these is a quadratic equation?

2

Methods of solving

Factorisation and the quadratic formula

Splitting the middle term; or the formula, which also gives irrational roots.

Factorisation

Split the middle term

Find two numbers with product $a c$ and sum $b$ .

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The quadratic formula

When it will not factorise

Identify $a, b, c$ , find $D = b^{2} - 4 a c$ , then $x = \frac{- b \pm D}{2 a}$ .

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3

Nature of the roots

The discriminant

D = b^{2} - 4 a c

The sign of $D$ tells whether the roots are real and when they are equal.

Deciding from $D$

Three cases and special cases

Know the nature without finding the roots; and $k$ questions.

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4

Relations among the roots

Sum, product and a new equation

Sum $- \frac{b}{a}$ and product $\frac{c}{a}$ ; expressions, forming equations and changing roots.

Expressions from sum and product

α + β

and

α β

Values like $α^{2} + β^{2}$ and $\frac{1}{α} + \frac{1}{β}$ without finding the roots.

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Forming equations and changing roots

x^{2} - S x + P = 0

The equation from given roots; and the new equation when the roots change.

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5

Word problems

Form the equation, then solve

Numbers, rectangles, speed-time and reciprocal numbers; and comparison questions with two equations.

Numbers and shapes

Pick the unknown, form the equation

Consecutive numbers, ratio and product, sum and product, and rectangles.

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Speed, reciprocals and comparison

Time gap,

x + \frac{1}{x}

,

x

versus

y

A quadratic from speed-time, number + reciprocal, and comparing $x$ and $y$ from two equations.

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6

Traps and fast tricks

Slips · tricks

Where marks go in quadratic-equation questions.

The commonest mistakes

Six traps

The slips that cost marks.

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Fast tricks

Sum-product and options

Habits for speed.

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7

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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8

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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The standard form

Recognising a quadratic

Methods of solving

Factorisation

The quadratic formula

Nature of the roots

Deciding from $D$

Relations among the roots

Expressions from sum and product

Forming equations and changing roots

Word problems

Numbers and shapes

Speed, reciprocals and comparison

Traps and fast tricks

The commonest mistakes

Fast tricks

The practice ladder

Level 1 — Basic

Level 2 — Standard

Level 3 — Advanced

Revision — formula cards

Free All Access Passevery mind map · Hindi + English · 12 months

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