What is Quadratic Equations Class 10: Definition & Concepts Explained

Definition and Standard Form of Quadratic Equations

A quadratic equation is a polynomial equation of degree 2 in the variable $x$. It can be written in the standard form:

$$ ax^2 + bx + c = 0 $$

where:

• $a$, $b$, and $c$ are real numbers
• $a \neq 0$ (if $a=0$, the equation becomes linear)

This form is the foundation for all methods used to solve quadratic equations in Class 10 NCERT Mathematics. Understanding this form helps you identify the coefficients and apply formulas correctly.

Methods to Solve Quadratic Equations in Class 10

There are three main methods to solve quadratic equations:

1. Factorisation Method

• Express the quadratic as a product of two binomials.
• Set each factor equal to zero to find roots.

2. Completing the Square

• Rewrite the quadratic in the form $(x + p)^2 = q$.
• Solve for $x$ by taking square roots.

3. Quadratic Formula

• Use the formula:

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

• This method works for all quadratic equations.

Example: Solve $x^2 - 5x + 6 = 0$ by factorisation.

• Factorise: $(x - 2)(x - 3) = 0$
• Roots: $x = 2$ or $x = 3$

These methods are part of the Class 10 NCERT syllabus and are essential for exam success.

Understanding the Discriminant and Nature of Roots

The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is given by:

$$ D = b^2 - 4ac $$

It helps determine the nature of the roots:

Knowing the discriminant helps you predict the solution type before solving the equation, a key skill in Class 10 exams.

Applications of Quadratic Equations in Real Life

Quadratic equations are not just theoretical; they model many real-world problems. Some common applications include:

• Projectile motion: The path of a thrown ball follows a quadratic equation.
• Area problems: Finding dimensions when area is given as a quadratic expression.
• Economics: Calculating profit maximisation or cost minimisation.

Example: The area of a rectangular garden is 120 m². If its length is 2 m more than its breadth, find the dimensions.

Let breadth = $x$ m, length = $x + 2$ m.

Area equation:

$$ x(x + 2) = 120 $$

Simplify:

$$ x^2 + 2x - 120 = 0 $$

Solve this quadratic to find $x$ and the length.

Comparing Methods: Factorisation vs Quadratic Formula

Choosing the right method to solve quadratic equations depends on the equation's form and complexity. Here's a comparison:

For Class 10 students, mastering all three methods is important for exam flexibility.

Important Formulas to Remember for Quadratic Equations

Here are key formulas every Class 10 student should memorize:

• Standard form: $ax^2 + bx + c = 0$
• Quadratic formula:

$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

• Discriminant: $D = b^2 - 4ac$
• Sum of roots: $\alpha + \beta = -\frac{b}{a}$
• Product of roots: $\alpha \times \beta = \frac{c}{a}$

These formulas are essential for solving problems efficiently and scoring well in exams.