Quadratic Equations Class 10 Notes, Concepts, Methods, Formulas & Solutions

Q: What is the simplest form of a quadratic?

$ax^2 + bx + c = 0$ without common factors and with simplified coefficients.

Q: How can a student quickly check correctness of roots?

Substitute them in the original equation and verify if both sides balance.

Q: What is the best revision strategy for this chapter?

Practice factorization, memorize formulas, and solve multiple word problems to gain confidence.

Q: Why do some equations need rearrangement before solving?

Because methods like factorization or formula application work only in standard form.

Q: What are exam-oriented word problems?

Problems involving numbers, age, geometry, motion, mixtures, and profit that reduce to quadratic equations.

Q: Why is discriminant analysis important in exams?

It quickly determines the nature of roots without solving the full equation.

Q: What is a real-life example of a quadratic equation?

Maximizing area of a rectangular garden using fixed fencing length leads to a quadratic equation.

Q: Does Class 10 require graphing of quadratics?

Only basic understanding; detailed graphing is taught in higher classes.

Q: What is the vertex form of a quadratic?

$y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Q: Can quadratic equations have irrational roots?

Yes, when $D$ is positive but not a perfect square.

QUADRATIC EQUATIONS-Notes

Quadratic Equations form one of the most important pillars of algebra in Class X Mathematics. This chapter introduces students to equations containing a squared variable, explores their real-life relevance, and teaches systematic methods to solve them. Whether it is calculating the area of a field, finding the height of a projectile, determining the dimensions of a rectangle, or solving number problems, quadratic equations appear everywhere in mathematics. This chapter aims to build conceptual clarity by explaining what makes an equation “quadratic”, how to convert real situations into quadratic form, and how to solve them using three highly structured methods.

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November 30, 2025 | By Academia Aeternum

QUADRATIC EQUATIONS-Notes

Maths - Notes

Quadratic Equations

A quadratic equation in variable $x$ is an equation of the form: \[ax^2+bx+c=0\] where,
$a,\ b,\ c$ are real numbers, $a\ne 0$.

Examples \[\begin{aligned}2x^2-3x=5=0,\\ x^2=0,\\ -3x^2+9x + 1=0\end{aligned}\]

Standard form of a quadratic equation.

Any equation of the form $p(x) = 0$, where $p(x)$ is a polynomial of degree 2, is a quadratic equation.
But when we write the terms of $p(x)$ in descending order of their degrees, then we get the standard form of the equation.
That is, \[ax^2 + bx + c = 0,\ a \ne 0\] is called the standard form of a quadratic equation.

When a Quadratic Equation Arises in Problems

Quadratic equations naturally appear in many situations:

Area and Geometry Problems
Examples includes:
- Finding the sides of a rectangle when area and relation between sides are given.
- Determining the diagonal of a square or dimensions of geometric shapes.
Number Problems
- Find two numbers whose sum is S and product is P
- If a number is increased by its reciprocal, result is K
Motion and Time Problems
- Speed-distance relations involving squared terms.
- Height of an object thrown upward.
Practical Value Problems
- Profit/Loss problems where cost and selling price appear in quadratic form.
- Revenue maximization.

Methods of Solving Quadratic Equations

Method of Factorization

Factorization is used when the quadratic can be broken into two linear factors.

General Idea:

Convert \[ax^2+bx+c=0\] into \[(ax+m)(x+n)=0\] and then set each factor equal to zero.

Steps:

Multiply $a\times c$
Split the middle term $b$ into two numbers whose product is $ac$.
Group and factor.
Use Zero-Product Property: \[\text{if } \\pq=0\\ \implies p=0 \text{ or } q=0\]

For Example:

\[ \begin{aligned} x^2+7x+10&=0\\ x^2+5x+2x+10&=0\\ x(x+5)+2(x+5)&=0\\ (x+2)(x+5)&=0\\ \implies \text{either } x+2&=0,\text{ or}\\ x+5&=0\\ \implies x=-2 \text{ or } x&=-5 \end{aligned} \]

Completing the Square

This method transforms the quadratic into a perfect square.

Idea

Rewrite \[ax^2+bx+c=0\] as \[(x+p)^2=q\]

Steps

Ensure coefficient of $x^2$ is 1.
Take half the coefficient of $x$, square it, and add to both sides.
Express left-hand side as a perfect square.
Take square roots on both sides.
Solve for $x$

This method is also the foundation for deriving the Quadratic Formula.

Quadratic Formula

This is the most universal method and works for every quadratic equation.
For Equation \[ax^2+bx+c=0\] the solutions are \[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]

Important Term: Discriminant (D)

\[D=b^2-4ac\] The discriminant tells us the nature of the roots.

Nature of Roots

$D\gt 0$: Roots are real and distinct

if $D\gt 0$ then \[x_1,\ x_2 = \frac{-b\pm\sqrt{D}}{2a}\]
$D=0$: Real and equal root

if $D = 0$ then \[x = \frac{-b}{2a}\]
$D\lt 0$: No real roots

if $D\lt 0$ then

the quadratic has no real solution (roots are complex).

Example-1

Check whether the following are quadratic equations:

$(x – 2)^2 + 1 = 2x – 3$
$(x + 2)^3 = x^3 – 4$

Solution

$$\begin{aligned}\left( x-2\right) ^{2}+1=2x-3\\ x^{2}+4-4x+1-2x+3=0\\ x^{2}-6x+8=0\end{aligned}$$ is in the standard form of Quadratic Equation $$\begin{aligned}ax^{2}+bx+c=0\\ \therefore \left( x-2\right) ^{2}+1=2x-3 \\\text{is a Quadratic Equation}\end{aligned}$$

$$\scriptsize\begin{aligned} (x+2) ^{3}=x^{3}-4\\ x^{3}+2^{3}+3x\cdot 2^{2}+3\cdot x^{2}.2-x^{3}+{4}=0\\ x^{3}+8+12x+6x^{2}-x^{3}+4=0\\ \Rightarrow 6x^{2}+12x+12=0\end{aligned}$$ is in the standard form of Quadratic Equation $$\begin{aligned}ax^{2}+bx+c=0\\ \therefore \left( x+2\right) ^{3}=x^{3}-4\\ \text{is a Quadratic Equation}\end{aligned}$$

Example-2

Find the roots of the equation $2x^2 – 5x + 3 = 0$, by factorisation.

Solution

$$2x^{2}-5x+3=0$$ By split middle term method, find two factors of $ac$ such that their algebraic sum is 5 $$\begin{aligned}2\times 3\\ \text{sum }\Rightarrow 2+3=5\end{aligned}$$ $$\begin{aligned}2x^{2}-2x-3x+3=0\\ 2x\left( x-1\right) -3\left( x-1\right) =0\\ \left( x-1\right) \left( 2x-3\right) =0\\ \Rightarrow \text{either }(x-1) =0\\ \text{or } \left( 2x-3\right) =0\\ \Rightarrow x=1\\ x=\dfrac{3}{2}\end{aligned}$$ Roots are $1$ and $\frac{3}{2}$

Example-3

Find the roots of the quadratic equation $6x^2 – x – 2 = 0$.

Solution

$$6x^{2}-x-2=0$$ Find the split middle term b factorising $ac$ such that algebraic sum of factors is 1 $$\begin{aligned}6\times 2\\ 4\times 3\\ 4-3=1\end{aligned}$$ $$\begin{aligned}6x^{2}-4x+3x-2=0\\ \Rightarrow 6x^{2}+3x-4x-2=0\\ 3x\left( 2x+1\right) -2\left( 2x+1\right) =0\\ \left( 2x+1\right) \left( 3x-2\right) =0\end{aligned}$$ ⇒ Either $$\begin{aligned}2x+1=0\\ x=-\dfrac{1}{2}\end{aligned}$$ or $$\begin{aligned}3x-2=0\\ x=\dfrac{2}{3}\end{aligned}$$ therefore roots are $$\text{Roots are }\boxed{\dfrac{1}{2},\dfrac{2}{3}}$$

Example-4

Find the roots of the quadratic equation $3x^2-2\sqrt{6}x+2$ .

Solution

$$\scriptsize\begin{aligned}3x^{2}-2\sqrt{6}x+2=0\\ \left( \sqrt{3}x\right) ^{2}-2\sqrt{3}\cdot \sqrt{2}x+\left( \sqrt{2}\right) ^{2}=0\\ a^{2}-2ab+b^{2}\\ =\left( a-b\right) \left( a-b\right) \\ \left( \sqrt{3}x-\sqrt{2}\right) \left( \sqrt{3}x-\sqrt{2}\right) =0\\ \sqrt{3}x-\sqrt{2}=0\\ \sqrt{3}x=\sqrt{2}\\ x=\dfrac{\sqrt{2}}{\sqrt{3}}\\ x=\sqrt{\dfrac{2}{3}}\end{aligned}$$

Roots are $\sqrt{\dfrac{2}{3}}$, $\sqrt{\dfrac{2}{3}}$

Example-5

Find the discriminant of the quadratic equation $2x^2 – 4x + 3 = 0$, and hence find the nature of its roots.

Solution

$$2x^{2}-4x+3=0$$ Discriminant $D$ $$\begin{aligned}D&=b^{2}-4ac\\ &=\left( -4\right) ^{2}-4\left( 2\times 3\right)\\ &=16-24\\ &=-8\end{aligned}$$ $D\lt 0$, Hence, Equation has no real roots

Example-6

Find the discriminant of the equation $3x^2 – 2x + \frac{1}{3}=0$ and hence find the nature of its roots. Find them, if they are real.

Solution

$$3x^{2}-2x+\dfrac{1}{3}=0$$

Multiplying both side of equation by 3

$$9x^{2}-6x+1=0$$

Discriminant

$$\begin{aligned}D&=b^{2}-4ac\\ &=\left( -6\right) ^{2}-4\left( 9\times 1\right) \\ &=36-36\\ &=0\end{aligned}$$

$D=0$, Hence, Equation has equal root

Finding the roots

$$\begin{aligned} \text{Roots }&=\dfrac{-b}{2a}\\ &=\dfrac{-\left( -6\right) }{2\times 9}\\ &=\dfrac{1}{3},\dfrac{1}{3}\end{aligned}$$

Frequently Asked Questions

A quadratic equation is an equation of the form $ax^2 + bx + c = 0$ where $a,\ b\, c$ are real numbers and $a \neq 0$.

If $a = 0$, the equation becomes linear and no longer contains a squared term, so it cannot be quadratic.

The standard form is $ax^2 + bx + c = 0$.

The word “quadratic” comes from “quad,” meaning square, because the highest power of the variable is 2.

The solutions of $ax^2 + bx + c = 0$ are $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

The discriminant $D$ is the expression $b^2 - 4ac$ found inside the square root of the quadratic formula.

It indicates two distinct real roots.

It indicates one real and repeated root.

It indicates no real roots; the solutions are complex.

By splitting the middle term into two terms whose product is (ac), factoring the expression, and using the zero-product property.

If $pq = 0$, then either $p = 0$ or $q = 0$. It is used to solve factored quadratic equations.

It means expressing $bx$ as the sum of two terms whose product equals $ac$, helping in factorization.

It is a method of rewriting a quadratic as a perfect square expression to solve the equation.

It helps derive the quadratic formula and solve equations that are not easy to factor.

Ensure $a = 1$, take half of the coefficient of $x$, square it, add it to both sides, form a perfect square, and solve.

Roots are the values of $x$ that satisfy the equation $ax^2 + bx + c = 0$.

For equation $ax^2 + bx + c = 0$: sum of roots = $-b/a$; product of roots = $c/a$.

The roots are the $x$-intercepts where the parabola $y = ax^2 + bx + c$ crosses the $x$-axis.

When $D = 0$; the parabola is tangent to the x-axis.

When $D < 0$; the graph does not cross or touch the $x$-axis.

It works for all types of quadratic equations, even when factorization is difficult.

They appear in geometry, projectile motion, business profit problems, age problems, and number-based puzzles.

Shape-based problems such as area, diagonal relations, and dimensions often result in a quadratic equation.

They are used in motion under gravity, height-time relations, and projectile trajectories.

By completing the square on the general form $ax^2 + bx + c = 0$.

An expression like $x^2 + 2px + p^2 = (x + p)^2$.

When its discriminant is a perfect square or when integers exist that multiply to $ac$ and sum to $b$.

Using relations involving area, speed, number constraints, or algebraic identities to form $ax^2 + bx + c = 0$.

If length = breadth + 3 and area = 40 sq units, then $b(b+3) = 40$ becomes a quadratic equation.

A quadratic equation without a linear term, i.e., of the form $ax^2 + c = 0$.

Quadratic equations that contain all three terms: $ax^2$, $bx$, and $c$.

A quadratic equation where $a = 1$, e.g., $x^2 + 5x + 6 = 0$.

Substituting answers back ensures the solution satisfies the original problem context.

Factorization-based questions, quadratic formula problems, word problems, discriminant evaluation, and root nature analysis.

Factorization is fastest when applicable; otherwise, the quadratic formula is the safest and most reliable.

Because the discriminant becomes negative, making the square root of a negative number impossible in real numbers.

Forgetting to bring the equation to standard form before applying methods or miscalculating the discriminant.

The sign determines the curve orientation and affects nature of roots.

Rearranging and simplifying the equation so that all terms are on one side of the equal sign.

Yes, when the discriminant is not a perfect square.