Quadratic Equations · Class 10 Mathematics: quick revision, 52 bites

The standard form

One shape for every quadratic

Takeaway. Everything in this chapter starts here.

The condition that matters

Why a cannot be zero

If $a = 0$ the $x^2$ term vanishes and $bx + c = 0$ is left — a linear equation.

Takeaway. No a, no square.

Appearances lie

Expand before you judge

$(x - 2)(x + 1) = (x - 1)(x + 3)$ looks quadratic.

Takeaway. Multiply the brackets out first, every time.

It can go the other way too

A cubic in disguise

Takeaway. Only expansion settles the degree.

Reading off a, b and c

Signs belong to coefficients

In $x^2 - 5x + 6 = 0$: $a = 1$, $b = -5$, $c = 6$.

Takeaway. The minus sign is part of b.

Still quadratic

Zeros are allowed — for b and c

$x^2 - 9 = 0$, $3x^2 + 5x = 0$ and $2x^2 = 0$ are all quadratic equations.

Takeaway. Only a is forbidden from being zero.

Substitute and get zero

What a root is

$\alpha$ is a root of $ax^2 + bx + c = 0$ when $a\alpha^2 + b\alpha + c = 0$.

Takeaway. Testing a root is one substitution.

Roots and zeroes

Two words, one set of numbers

The roots of the EQUATION $ax^2 + bx + c = 0$ are the zeroes of the POLYNOMIAL $ax^2 + bx + c$.

Takeaway. Chapter 2's zeroes, under a new name.

How many roots

At most two

A quadratic polynomial has at most two zeroes, so a quadratic equation has at most two roots — sometimes none at all among real numbers.

Takeaway. Never three.

Forming an equation

Three moves, every word problem

Takeaway. The last step carries its own mark.

Rectangle problems

Areas make squares

Takeaway. Breadth x, length 2x + 1, area 528 → $2x^2 + x - 528 = 0$.

Consecutive integers

One apart or two?

Consecutive integers are x and $x + 1$.

Takeaway. Read the word before 'integers'.

Age problems

Everyone ages together

'Three years from now' adds 3 to BOTH people's ages.

Takeaway. Adjust every age in the problem.

Speed and time

Fractions become squares

Takeaway. $\frac{480}{x-8} - \frac{480}{x} = 3$ becomes $x^2 - 8x - 1280 = 0$.

The last mark

Finish the simplification

A 'represent the situation' answer is not complete until the equation is in standard form, with everything on one side.

Takeaway. Stopping at the fraction equation loses a mark.

Is it a root?

How to test a given number

Takeaway. No solving required — just arithmetic.

Where quadratics come from

Older than algebra itself

Babylonian clay tablets from around 1800 BCE set problems that amount to solving quadratic equations, long before symbols existed.

Takeaway. Brahmagupta gave an explicit rule in 628 CE.

The zero-product rule

Zero is special

If $pq = 0$ then $p = 0$ or $q = 0$.

Takeaway. One equation becomes two.

Multiply to ac, add to b

The split you are looking for

Takeaway. A pair that fails either test will not factorise.

2x² − 5x + 3 = 0

A full factorisation

Takeaway. The repeated bracket shows the split was right.

The bracket must repeat

Your own error check

After grouping, the two brackets must be IDENTICAL.

Takeaway. No repeated bracket, no factorisation.

Reading the root off a factor

The sign flips

The factor $(x - 2)$ gives the root $+2$, not $-2$.

Takeaway. Solve the bracket, don't read it.

Common factors first

Divide before you hunt

$2x^2 - 14x - 120 = 0$ becomes $x^2 - 7x - 60 = 0$.

Takeaway. One division saves half the work.

Equal roots

Two roots, both the same

$x^2 + 4x + 4 = 0$ factorises as $(x + 2)^2 = 0$.

Takeaway. A perfect square gives equal roots.

Seven steps

The method end to end

Takeaway. Step 6 is where the marks are lost.

Why the right side matters

Only zero will do

The rule applies to $pq = 0$, never to $pq = 6$.

Takeaway. Bring it to zero before you factorise.

6x² − x − 2 = 0

Opposite signs

Takeaway. A negative ac means the pair has opposite signs.

Rejecting a root

Correct but wrong

Both roots solve the equation; only one may fit the story.

Takeaway. Write the rejection down — it carries a mark.

Product 306

Two consecutive integers

$x^2 + x - 306 = 0$ factorises as $(x + 18)(x - 17) = 0$.

Takeaway. The answer is 17 and 18.

480 km, 8 km/h slower, 3 hours more

The train

$x^2 - 8x - 1280 = 0$ gives $(x - 40)(x + 32) = 0$.

Takeaway. Reject, then answer in words.

The product is what counts

Not a and c separately

For $2x^2 - 5x + 3$, use $ac = 6$ — not 2 and 3 on their own.

Takeaway. One number, ac, and one number, b.

Always verify

Prove it to yourself

Substitute one root into the ORIGINAL equation before moving on. It costs ten seconds and catches sign errors.

Takeaway. The original, not your factorised form.

When to stop trying

Some do not split

$x^2 + x - 1 = 0$ has no integer pair multiplying to $-1$ and adding to 1.

Takeaway. Failure to factorise is not failure to have roots.

Where the method comes from

Older than the symbols

Bhaskara II, writing in 1150 CE, solved quadratic equations by methods close to these.

Takeaway. The algebra is newer than the mathematics.

The discriminant

One number, three answers

Takeaway. It names the case without solving anything.

The three cases

What the sign means

Takeaway. Learn this table and half the topic is done.

The quadratic formula

When roots exist, this finds them

Takeaway. It works on every quadratic, not only the factorisable ones.

The squaring trap

b² is never negative

If $b = -5$ then $b^2 = 25$.

Takeaway. Square first, then think about signs.

Signs in −4ac

Two negatives, one plus

For $2x^2 + x - 3 = 0$: $-4ac = -4(2)(-3) = +24$, so $D = 25$.

Takeaway. A negative c usually makes D bigger, not smaller.

What 2a divides

The whole numerator

$x = \dfrac{-b \pm \sqrt{D}}{2a}$ — the $-b$ is divided too.

Takeaway. One fraction bar, over everything.

−b in the formula

The sign of b flips

If $b = -5$ then $-b = +5$.

Takeaway. Substitute b with its sign, then negate.

3x² − 5x + 2 = 0

A full solution

Takeaway. D first, formula second.

Two equal roots, not one

Say it the right way

$D = 0$ gives TWO EQUAL roots — the factor appears twice.

Takeaway. The wording itself carries marks.

Perfect squares

Was factorisation ever possible?

If $D > 0$ AND is a perfect square, the roots are rational and the equation factorises.

Takeaway. D quietly tells you which method to use.

Irrational roots

Why x² + x − 1 would not split

$D = 1 + 4 = 5$, not a perfect square.

Takeaway. No integer split could have found these.

Finding a parameter k

Equal roots means D = 0

Takeaway. The last step rules out k = 0.

2x² + kx + 3 = 0 with equal roots

Both signs count

$k^2 - 24 = 0$ gives $k^2 = 24$, so $k = \pm 2\sqrt6$.

Takeaway. A squared unknown has two answers.

The k = 0 trap

Keep it quadratic

For $kx^2 - 2kx + 6 = 0$, $D = 0$ gives $k = 0$ or $k = 6$.

Takeaway. Reject a k that destroys the $x^2$ term.

Is the design possible?

A negative D is a real answer

A park of perimeter 80 m and area 500 m² gives $D = -400$.

Takeaway. The discriminant is the argument.

Perimeter 80 m, area 400 m²

The one that just works

$x^2 - 40x + 400 = 0$ gives $D = 0$.

Takeaway. D = 0 means exactly one shape fits.

Answering 'nature of roots'

Three lines, three marks

Write a, b and c; compute D showing the arithmetic; state the conclusion in words.

Takeaway. The conclusion in words is a mark of its own.

Discriminant

Why it is called that

The word means 'the thing that distinguishes' — it separates the three cases without solving anything.

Takeaway. When D < 0 the roots are complex, which is Class 11's story.