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Quadratic Equations Class 10 ICSE Maths Revision Notes Chapter 5 PDF

Quadratic Equations Class 10 ICSE Maths Revision Notes Chapter 5

Hi students, Welcome to Amans Maths Blogs (AMB). In this article, you will get Quadratic Equations Class 10 Maths Revision Notes Chapter 5 PDF.

Quadratic Polynomials

A polynomial of the form of p(x) = ax2 + bx + c is known as quadratic polynomial, where a, b, c are real and a ≠ 0.

For example: p(x) = 3x2 + 5x – 2, here a = 3, b = 5 and c = -2.

Quadratic Equations

An equation in the form of ax2 + bx + c = 0 is known as quadratic equation, where a, b, c are real and a ≠ 0.

For example: 3x2 + 5x – 2 = 0, here a = 3, b = 5 and c = -2.

Solving Quadratic Equations by Factorization

To solve a quadratic equation by factorization, we do the following steps:

Step 1 : Express the equation in the form of ax2 + bx + c = 0.

Step 2 : Factorize ax2 + bx + c using middle term splitting method.

Step 3 : Put each factor equal to zero.

Step 4 : Solve each equation.

Equations Reducible to Quadratic Equations

An equation in the form of ax2 + bx + c = 0 is known as quadratic equation, where a, b, c are real and a ≠ 0.

In some questions, the given equations are not in form of ax2 + bx + c = 0. Then, to solve such type of equations, first we need to reduce the given equation in form of quadratic equation ax2 + bx + c = 0 by using suitable algebraic transformation and then solve it.

Solving Quadratic Equation by Quadratic Formula

An equation in the form of ax2 + bx + c = 0 is known as quadratic equation, where a, b, c are real and a ≠ 0.

To solve quadratic equations, there are two methods :

1. Using Factorization Method

2. Using Quadratic Formula

In this exercise, the quadratic equations are solved by using the quadratic formula.

solving quadratic equations by quadratic formula amans maths blogs

Proof of Quadratic Formula

Let the quadratic equation is ax2 + bx + c = 0, where a ≠ 0. We need to find the value of x.

ax2 + bx + c = 0

ax2 + bx = – c [Transposing constant term c to RHS]

x2 + (b/a)x = – c/a [Dividing both sides by a to make coefficient of x2 = 1]

x2 + (b/a)x + b2/4a2 = b2/4a2 – c/a [Adding both sides by b2/4a2]

x2 + 2(b/2a)x + b2/4a2 = (b2 – 4ac)/4a2

⇒ (x + b/2a)2 = (b2 – 4ac)/4a2

⇒ (x + b/2a) = ± √(b2 – 4ac)/2a

x = – b/2a ± √(b2 – 4ac)/2a

x = – b ± √(b2 – 4ac)]/2a.

Therefore, if the quadratic equation ax2 + bx + c = 0, then the value of x is x = [- b ± √(b2 – 4ac)]/2a. This is known as Quadratic Formula.

Hence, there are two values of x as x = [- b + √(b2 – 4ac)]/2a or x = [- b + √(b2 – 4ac)]/2a.

Discriminant of Quadratic Equations

An equation in the form of ax2 + bx + c = 0 is known as quadratic equation, where a, b, c are real and a ≠ 0. 

Then the value of x is x = [- b ± √(b2 – 4ac)]/2a. This is known as Quadratic Formula. 

Hence, there are two values of x as x = [- b + √(b2 – 4ac)]/2a or x = [- b + √(b2 – 4ac)]/2a.

Now, in x = [- b ± √(b2 – 4ac)]/2a, the term under square root, that is (b2 – 4ac) is known as the discriminant of the quadratic equation of ax2 + bx + c = 0 and it is denoted by D.

Thus, D = b2 – 4ac.

discriminant of quadratic equation amans maths blogs

Nature of Roots of Quadratic Equations

The nature of the roots determines the types of the roots of quadratic equation of ax2 + bx + c = 0, whether the roots are real or imaginary and if the roots are real, then it will be rational or irrational and equal or unequal. 

Since D = b2 – 4ac is in quadratic formula x = [- b ± √(b2 – 4ac)]/2a, then the value of the discriminant D = b2 – 4ac determines the nature of the roots of quadratic equations.

You don’t need to solve the quadratic equation of ax2 + bx + c = 0. You just need to find the value of its discriminant D = b2 – 4ac.

How To Determine Nature of Roots of Quadratic Equations

If the discriminant D = b2 – 4ac is positive (means, D > 0), then the roots of quadratic equation are REAL and DISTINCT.

If the discriminant D = b2 – 4ac is positive and perfect square, then the roots of quadratic equation are REAL, RATIONAL and DISTINCT

If the discriminant D = b2 – 4ac is positive and NOT a perfect square, then the roots of quadratic equation are REAL, IRRATIONAL and DISTINCT.

If the discriminant D = b2 – 4ac is zero (means, D = 0), then the roots of quadratic equation are REAL and EQUAL. And, Root will be α = β = -b/2a.

If the discriminant D = b2 – 4ac is negative (means, D < 0), then the roots of quadratic equation are  NOT REAL , it means the roots are IMAGINARY.

Summary of the nature of the roots of quadratic equation is as below.

how to determine nature of roots of quadratic equations class 10 maths amans maths blogs

Quadratic Equations Word Problems

In word problems, first we need to make a quadratic equation according to the given question and then we need to solve them.

Read : S Chand Class 10 ICSE Maths Quadratic Equations Exercise 5A Solutions PDF

Read : S Chand Class 10 ICSE Maths Quadratic Equations Exercise 5B Solutions PDF

Read : S Chand Class 10 ICSE Maths Quadratic Equations Exercise 5C Solutions PDF

Read : S Chand Class 10 ICSE Maths Quadratic Equations Exercise 5D Solutions PDF

Read : S Chand Class 10 ICSE Maths Quadratic Equations Exercise 5E Solutions PDF

Read : S Chand Class 10 ICSE Maths Quadratic Equations Revision Exercise Solutions PDF

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