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Quadratic Equations Class 10 Maths Notes Chapter 4

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CBSE Class 10 Maths Notes Chapter 4 Quadratic Equations are an exceptional resource created by our team of experienced Subject Experts at GfG. As staunch supporters of students’ education, we prioritize their learning and development above all else, and that is why we have created these comprehensive notes to help them better understand the complex topic of Quadratic Equations.

Chapter 4 of the NCERT Class 10 Maths textbook delves into the world of Quadratic Equations and covers various topics such as finding the roots of quadratic equations, graphical representations, discriminant, nature of roots, and many others. These notes are designed to provide students with a comprehensive summary of the entire chapter and include all the essential topics, formulae, and concepts needed to succeed in their exams.

What is Quadratic Equation?

An algebraic equation of the second degree is called the quadratic equation i.e., an equation in one variable with the highest possible term containing a square term of the variable.

A quadratic equation in variable “x” is an equation of the form, 

ax2 + bx + c = 0

Where, a, b, and c are real numbers and a ≠ 0.

Some examples of quadratic equations are,

  • 3x2 – 11x + 23 = 0
  • 5x2 = 0
  • 11x2 -13 = 0

Generally, any second-degree polynomial P(x), when put like P(x) = 0 represents a quadratic equation. 

Standard Form of a Quadratic Equation

The standard form of quadratic equations is,

ax2 + bx + c = 0

Where,

  • x is the variable of the equation,
  • a, b, and c are real numbers and constants and a ≠ 0

Roots of a Quadratic Equation

The values of the x0 which satisfies the quadratic equation q(x) are called the roots of the quadratic equation. This implies that for any xo if

q(x0) = 0, then x0 is the root of the q(x).

As the quadratic equation has two degrees it can have a maximum of 2 roots.

Graph of Quadratic Equation

The graph of a quadratic equation is always parabolic, where its shape depends upon the coefficient of x2 i.e., “a” in the general equation of quadratic. If a>0, the parabola opens upward, and if a < 0, the parabola opens downward. Also, the point at which the graph intersects the x-axis is its root.

The image given below shows the Quadratic Equation which opens upwards.

Quadratics equation which opens upward

 

The image given below shows the Quadratic Equation which opens downwards.

Quadratics equation which opens upward

 

Another factor that affects the position of the graph is Discriminant, 

  • D > 0, there are two roots of the quadratic equation i.e., it intersects the x-axis at two points.
  • If D = 0, then there is only one root i.e., the graph touches the x-axis, and 
  • If D < 0, then there is no real root of the quadratic equation i.e., the graph never cuts the x-axis. It’s either above the x-axis or below the x-axis.
Graphs of three quadratics equations, for each condition of D.

 

Methods for Solving Quadratic Equations

Let’s assume a quadratic equation P(x) = 0. The points which satisfy this equation are called solutions or zeros of this quadratic equation. There are three types of methods to find the solution of a quadratic equation: 

  • Factorization Method
  • Completing the Squares Method
  • Shree Dharacharya or Quadratic Formula

By Factorization Method

A quadratic equation can be considered a factor of two terms. Like ax2 + bx + c = 0 can be written as (x – x1)(x – x2) = 0 where x1 and x2 are roots of quadratic equation.

Step 1: Find two numbers such that the product of the numbers is ‘ac’ and the sum is ‘b’.

Step 2:Then write x coefficient as the sum of these two numbers and split them such that you get two terms for x.

Step 3: Factor the first two as a group and the last two terms as another group.

Step 4: Take common factors from these and on equating the two expressions with zero after taking common factors and rearranging the equation we get the roots.

Example: Find the roots of the following quadratic equation using the factorization method. 

2x2 – x – 6 = 0

Solution: 

2x2 – x – 6 = 0

⇒ 2x2 – 4x. +3x – 6 = 0

⇒ 2x (x – 2) +3(x – 2) = 0 

⇒ (2x + 3) (x – 2) = 0 

For, 2x + 3 = 0 

⇒  x = -3/2

For, x – 2 = 0 

⇒  x = 2

Thus, this equation has roots x = 2 and -3/2

By Completing the Square Method

Any equation ax2 + bx + c = 0 can be converted in the form (x + m)2 – n2 = 0. After this take the square roots and get the roots of the equation. Completing the square is just a way to readjust the given quadratic equation so that they come in the form of complete squares. Let’s see this through an example.

Example 1: Find the root of the given equation by completing the square method. 

x2 + 4x – 5 = 0

Solution:

We are given, x2 + 4x – 5 = 0 

To solve it by completing the square method, we need to bring it in the above mentioned form. 

x2 + 4x – 5 = 0

⇒ x2 + 4x + 4 – 9 = 0

⇒ (x + 2)2 – 32 = 0

⇒ (x + 2)2 = 32

Taking square root both sides, 

x + 2 = 3 and x + 2 = -3 

This gives us x = 1, -5

By Quadratic Formula Method

This formula says, 

For a quadratic equation in general form, ax2 + bx + c = 0

If b2 – 4ac > 0, 

Then its roots are given by \frac{-b\pm \sqrt{b^2-4ac}}{2a}

Example: Find the roots of equation 3x2 – 5x + 2 = 0.

Solution: 

For finding out the roots using Shree Dharacharya formula, 

We need to check If b2 – 4ac > 0,

In this particular equation, a = 3, b = -5 and c = 2. 

So, b2 – 4ac 

⇒ (-5)2 – 4(3)(2) 

⇒ 25 – 24 

⇒ 1 > 0 

Thus, roots are possible, 

Now let’s calculate the roots by plugging in the values in the formula mentioned above. 

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

=\frac{-5\pm \sqrt{5^2-4(3)(2)}}{2(3)}

=\frac{-5\pm1}{6}

=\frac{-4}{6},\frac{-6}{6}

=\frac{-2}{3},-1

Discriminant

The discriminant of the quadratic equation ax2 + bx + c = 0 is calculated using the formula,

D(Discriminant) = b2 – 4ac

The discriminant of the quadratic equation helps us find the nature of the roots which is explained in the next heading.

Nature of Roots

To find the nature of the roots of the quadratic equation we find the discriminant of the given quadratic equation. The term is called discriminant because it determines the nature of the roots of the quadratic equation based on its sign.

There are 3 types in the nature of roots, 

  • Real and distinct roots: For real and distinct roots, the discriminant should be positive i.e. b2 – 4ac > 0. The curve of the equation intersects the x-axis at two different points.
  • Real and equal roots: For real and equal roots, the discriminant is zero i.e. b2 – 4ac = 0. The curve of the equation intersects the x-axis at only one point.
  • Complex roots: For complex roots, the discriminant is negative i.e. b2 – 4ac = 0. The curve of the equation does not intersect the x-axis.

Relation between Coefficient and Roots

If f(x) = ax2 + bx + c is a quadratic equation with roots α and β then,

  • Sum of roots = α + β = -b/a
  • Product of roots = αβ = c/a

Formation of Quadratic Equation from Roots

If the sum and product of roots are given then, the quadratic equation is given by :

  • x2 – (sum of roots)x + (product of roots) = 0

If roots are α and β, then the quadratic equation becomes,

  • x2 – (α + β)x + α × β = 0

Some other Important Things to Remember:

  • The quadratic equations a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0 have;
    • One common root if (b1c2 – b2c1)/(c1a2 – c2a1) = (c1a2 – c2a1)/(a1b2 – a2b1)
    • Both roots common if a1/a2 = b1/b2 = c1/c2
  • In quadratic equation ax+ bx + c = 0 or [(x + b/2a)2 – D/4a2]
    • If a > 0, minimum value = 4ac – b2/4a at x = -b/2a.
    • If a < 0, maximum value 4ac – b2/4a at x= -b/2a.
  • If α, β, γ are roots of the cubic equation ax3 + bx2 + cx + d = 0, then, α + β + γ = -b/a, αβ + βγ + λα = c/a, and αβγ = -d/a

The image added below shows the condition when the a > 0 and  a < 0 for the quadratic equation ax2 + bx + c =0

 a is greater than 0 and  a less than 0 conditions

 

Related Resources

FAQs on NCERT Notes for Class 10 Maths Chapter 4 Quadratic Equation

Q1: What is a quadratic equation?

Answer:

Quadratic equation is second degree equation of any variable with the general form ax2 + bx + c = 0, where a, b, and c are real number and a ≠ 0.

Q2: How do you solve a quadratic equation?

Answer:

There are various methods to solve quadratic equation, some of which are as follows:

  • Factorization Method
  • Completing the Squares Method
  • Shree Dharacharya or Quadratic Formula

Q3: What is the quadratic formula?

Answer:

The quadratic formula is a formula used to solve quadratic equations. For a quadratic equation in general form, ax2 + bx + c = 0

Quadratic formula is given by,

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

Where, a, b, and c are real number and a ≠ 0.

Q4: What are the roots of a quadratic equation?

Answer:

A real value xo which satisfy the quadratic expression f(x) such that f(xo) = 0, is called root of the quadratic equation and a quadratic equation can have at most 2 roots.

Q5: What is the discriminant of a quadratic equation?

Answer:

 For a quadratic equation in general form, ax2 + bx + c = 0

Discriminant is given by

D = b2 – 4ac



Last Updated : 18 Apr, 2023
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