Quadratic Equations for Class 10: Everything You Need to Score Full Marks

Why Quadratic Equations Are a Board Exam Favourite

Quadratic equations appear in CBSE and ICSE board exams year after year without exception. The chapter carries approximately 5 to 6 marks, but its real importance goes beyond the direct marks. Quadratic thinking shows up in word problems from other chapters too, especially when you are finding areas, ages, or speeds.

The good news is that unlike trigonometry proofs or geometric constructions, quadratic equations follow clear, systematic methods. Once you learn the three approaches and understand when to use each one, you can solve any quadratic equation the board throws at you.

Let us start from the absolute basics and build up to the exam level problems that trip students up the most.

What Is a Quadratic Equation?

The 3 Methods of Solving Quadratic Equations

The Discriminant: Nature of Roots

Conquering Word Problems

Word problems involving quadratic equations are feared by most students, yet they carry 5 marks in board exams. The key is a systematic translation process.

Step 1: Read the problem twice. Identify what is unknown and assign it a variable (let the number be $x$, let the age be $x$, let the speed be $x$).

Step 2: Translate the English sentences into mathematical relationships. "A number and its square add up to 42" becomes $x + x^2 = 42$, which rearranges to $x^2 + x - 42 = 0$.

Step 3: Solve the quadratic equation using any method.

Step 4: Check both roots against the original problem. Often one root is mathematically valid but does not make sense in context (like a negative age or speed). Reject it and state why.

Common word problem types: number problems (find two numbers whose sum and product are given), age problems (present and past/future ages), speed and distance (total time for a journey), and area problems (dimensions of a rectangle given perimeter and area).

5 Solved Examples

Example 1: Factorisation
Solve $x^2 - 7x + 12 = 0$
Find two numbers with sum 7 and product 12: that is 3 and 4.
$x^2 - 3x - 4x + 12 = 0 \Rightarrow x(x - 3) - 4(x - 3) = 0 \Rightarrow (x - 3)(x - 4) = 0$
Roots: $x = 3$ or $x = 4$

Example 2: Quadratic Formula
Solve $2x^2 + 3x - 5 = 0$
Here $a = 2, b = 3, c = -5$. Discriminant $D = 9 + 40 = 49$.
$x = \frac{-3 \pm 7}{4}$, so $x = 1$ or $x = \frac{-10}{4} = -2.5$

Example 3: Nature of Roots
For what value of $k$ does $x^2 + kx + 9 = 0$ have equal roots?
For equal roots, $D = 0$: $k^2 - 36 = 0$, so $k = \pm 6$

Example 4: Word Problem
The product of two consecutive positive integers is 272. Find them.
Let the integers be $x$ and $x + 1$. Then $x(x + 1) = 272$, giving $x^2 + x - 272 = 0$.
Using the formula: $x = \frac{-1 + \sqrt{1 + 1088}}{2} = \frac{-1 + 33}{2} = 16$.
The integers are 16 and 17. (Reject $x = -17$ since they must be positive.)

Example 5: Completing the Square
Solve $x^2 + 4x - 5 = 0$ by completing the square.
$x^2 + 4x = 5$. Add $(4/2)^2 = 4$ to both sides: $x^2 + 4x + 4 = 9$.
$(x + 2)^2 = 9$, so $x + 2 = \pm 3$. Therefore $x = 1$ or $x = -5$.

The 7 Mistakes That Cost Students Marks

How SparkEd Breaks Down Quadratic Equations Visually

At SparkEd, every quadratic equation is solved with a visual step by step walkthrough. You do not just see the factorisation or the formula application. You see why each step follows from the previous one.

For factorisation problems, SparkEd shows the factor pair search visually, highlighting which pairs work and which do not. For the quadratic formula, each computation ($b^2$, then $4ac$, then $D$, then $\sqrt{D}$, then the final division) is shown as a separate labelled step.

For word problems, SparkEd walks you through the translation from English to algebra, then solves the equation, and finally verifies the answer against the original problem context.

With three difficulty levels, you can start with straightforward factorisation (Easy), move to discriminant and formula questions (Medium), and then tackle multi step word problems (Hard). Our AI coach Spark helps you through stuck moments without giving away the answer.

Written by the SparkEd Math Team

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