Quadratic Equations Class 10 ICSE Maths Revision Notes Chapter 5 PDF

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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.

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.

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.

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.

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