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Quadratic Equation Class 10 Notes: Chapter 4

Get the complete concepts covered in quadratic equations for class 10 Maths here. In this article, you will learn the concept of quadratic equations, standard form, nature of roots, methods for finding the solution for the given quadratic equations with more examples.

Introduction to Quadratic Equations

Quadratic Polynomial

A polynomial of the form $x^2+bx+c$, where a,b and c are real numbers and $a\ne 0$ is called a quadratic polynomial.

For More Information On Quadratic Polynomials, Watch The Below Video.

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Quadratic Equation

When we equate a quadratic polynomial to a constant, we get a quadratic equation.

Any equation of the form p(x)=c, where p(x) is a polynomial of degree 2 and c is a constant, is a quadratic equation.

The standard form of a Quadratic Equation

The standard form of a quadratic equation is $a{x}^2+bx+c=0$, where a,b and c are real numbers and $a\ne 0$.
‘a’ is the coefficient of $x^2$. It is called the quadratic coefficient. ‘b’ is the coefficient of x. It is called the linear coefficient. ‘c’ is the constant term.

Solving QE by Factorisation

Roots of a Quadratic equation

The values of $x$ for which a quadratic equation is satisfied are called the roots of the quadratic equation.

If α is a root of the quadratic equation $a{x}^2+bx+c=0$, then$a{\alpha }^2+b\alpha +c=0.$

A quadratic equation can have two distinct real roots, two equal roots or real roots may not exist.

Graphically, the roots of a quadratic equation are the points where the graph of the quadratic polynomial cuts the x-axis.

Consider the graph of a quadratic equation $x^2-4=\mathrm{0:}$

Graph of a Quadratic Equation

In the above figure, -2 and 2 are the roots of the quadratic equation $x^2-4=0$
Note:

• If the graph of the quadratic polynomial cuts the x-axis at two distinct points, then it has real and distinct roots.
• If the graph of the quadratic polynomial touches the x-axis, then it has real and equal roots.
• If the graph of the quadratic polynomial does not cut or touch the x-axis then it does not have any real roots.

Solving a Quadratic Equation by Factorization method

Consider a quadratic equation $2x^2-5x+3=0$
$\Rightarrow 2x^2-2x-3x+3=0$
This step is splitting the middle term
We split the middle term by finding two numbers (-2 and -3) such that their sum is equal to the coefficient of $x$ and their product is equal to the product of the coefficient of $x^2$ and the constant.
(-2) + (-3) = (-5)
And (-2) $\times$ (-3) = 6
$2x^2-2x-3x+3=0$
$2x(x-1)-3(x-1)=0$
$(x-1)(2x-3)=0$
In this step, we have expressed the quadratic polynomial as a product of its factors.
Thus, x = 1 and x =3/2 are the roots of the given quadratic equation.
This method of solving a quadratic equation is called the factorisation method.

For More Information On Solving a Quadratic Equation by Factorization method, Watch The Below Video.

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