The standard form
One shape for every quadratic
↳ Everything in this chapter starts here.
📚 Chapter Learning Kit
Quadratic Equations · Class 10 Mathematics
English 3 topics 170 leveled MCQs 60 flashcards 3 games Free
Shared by a Veda teacher · Generated with Veda AI · Sep 2026
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💡 Key Idea
⚠️ Common Mistake
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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.
↳ No a, no square.
Appearances lie
Expand before you judge
$(x - 2)(x + 1) = (x - 1)(x + 3)$ looks quadratic.
↳ Multiply the brackets out first, every time.
The zero-product rule
Zero is special
If $pq = 0$ then $p = 0$ or $q = 0$.
↳ One equation becomes two.
Multiply to ac, add to b
The split you are looking for
↳ A pair that fails either test will not factorise.
2x² − 5x + 3 = 0
A full factorisation
↳ The repeated bracket shows the split was right.
📖 Smart notes
What you'll study, topic by topic
1
What a Quadratic Equation Is
A quadratic equation is any equation that can be written as $ax^2 + bx + c = 0$ with $a \ne 0$. Deciding whether an equation is quadratic means expanding it first, and many real situations — areas, consecutive numbers, a…
- The standard form is $ax^2 + bx + c = 0$, where a, b, c are real numbers and $a \ne 0$.
- If $a = 0$ the equation is linear, not quadratic — that is the whole point of the condition.
- Never judge by appearance: EXPAND and simplify first, then look at the highest power.
~40 min · full explanation, examples & memory tricks in the app
2
Solving by Factorisation
Split the middle term into two pieces that multiply to $ac$ and add to $b$, factorise into two brackets, then set each bracket to zero. If a product is zero, one of its factors must be zero — that single fact is what tur…
- ZERO-PRODUCT RULE: if $pq = 0$ then $p = 0$ or $q = 0$ — this is why factorisation solves equations.
- Always bring the equation to standard form $ax^2 + bx + c = 0$ before factorising.
- To split the middle term, find two numbers whose PRODUCT is $ac$ and whose SUM is $b$.
~45 min · full explanation, examples & memory tricks in the app
3
The Discriminant and the Quadratic Formula
The single number $b^2 - 4ac$ tells you, before any solving, whether an equation has two distinct roots, two equal roots, or none at all. When roots exist, the quadratic formula produces them — and it works on every quad…
- The DISCRIMINANT of $ax^2 + bx + c = 0$ is $D = b^2 - 4ac$.
- If $D > 0$: two distinct real roots.
- If $D = 0$: two equal real roots, both $-\frac{b}{2a}$.
~45 min · full explanation, examples & memory tricks in the app
❓ Leveled MCQ practice
Try the smart MCQs from this kit
170 questions laddered from warm-up to topper-level, each with an explanation. A taste:
The standard form of a quadratic equation is
Beginner
A $ax^2 + bx + c = 0,\ a = 0$
B $ax + b = 0$
C $ax^2 + bx + c = 0,\ a \ne 0$
D $ax^3 + bx^2 + c = 0$
Show answer & explanation
$ax^2 + bx + c = 0,\ a \ne 0$
Degree two requires a non-zero coefficient of $x^2$.
Setting $a = 0$ in $ax^2 + bx + c = 0$ turns it into
Beginner
A a cubic equation
B a linear equation
C a quadratic equation still
D no equation at all
Show answer & explanation
a linear equation
Without the $x^2$ term only $bx + c = 0$ is left, which has degree one.
Which of these is a quadratic equation?
Beginner
A $3x - 7 = 0$
B $x^3 + 1 = 0$
C $\frac{1}{x} = 4$
D $x^2 - 9 = 0$
Show answer & explanation
$x^2 - 9 = 0$
Only the first has highest power 2 with a non-zero coefficient.
In $x^2 - 5x + 6 = 0$, the value of b is
Beginner
A 5
B −5
C 6
D 1
Show answer & explanation
−5
The coefficient of x carries its sign.
🃏 Flashcards
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