CBSE Grade 10 Maths Chapter 4 - Quadratic Equation

Introduction to Quadratic Equations

Quadratic Polynomial

A polynomial, whose degree is 2, is called a quadratic polynomial.
It is in the form of

p(x) = ax²+ bx + c

where a,b and c are real numbers and a≠0

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

ax²+bx+c =0

where a,b and c are real numbers and a≠0

‘a’ is the coefficient of x² (It is called the quadratic coefficient.)
‘b’ is the coefficient of x. (It is called the linear coefficient).
c is a constant term.

Types of Quadratic Equations

1. Complete Quadratic Equation ax²+ bx + c = 0, where a ≠ 0, b ≠ 0, c ≠ 0

2. Pure Quadratic Equation ax²= 0, where a ≠ 0, b = 0, c = 0

Roots of a Quadratic Equation

Let x = α where α is a real number.

If α satisfies the Quadratic Equation ax²+ bx + c = 0 such that aα²+ bα + c = 0, then α is the root of the Quadratic Equation.

As quadratic polynomials have degree 2, therefore Quadratic Equations can have two roots.

So the zeros of quadratic polynomial p(x) =ax²+bx+c is same as the roots of the Quadratic Equation ax²+ bx + 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 x²-9

In the above figure, -3 and 3 are the roots of the quadratic equation x²-9 = 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 Quadratic Equation by Factorisation Method
In this method, we factorise the equation into two linear factors and equate each factor to zero to find the roots of the given equation.

Step 1: Given Quadratic Equation in the form of ax²+ bx + c = 0.

Step 2: Split the middle term bx as mx + nx so that the sum of m and n is equal to b and the product of m and n is equal to ac.

Step 3: By factorization we get the two linear factors (x + p) and (x + q)

ax²+ bx + c = 0 = (x + p) (x + q) = 0

Step 4: Now we have to equate each factor to zero to find the value of x.

Example : Solve quadratic equation 2x²+7x+3.

Solution :

Step 1: Write value of ac and b.

ac = 2*3 = 6 and b = 7

Step 2 : Find two numbers that multiply together to make 6, and add up to 7

In fact 6 and 1 do that (6×1=6, and 6+1=7)

Step 3: Rewrite the middle term with those numbers:

Rewrite 7x as sum of 6x and 1x

2x² + 6x + x + 3

Step 4: Factor the first two and last two terms separately

The first two terms 2x²+ 6x factor into 2x(x+3)

The last two terms x+3 don't actually change in this case.

So we get:

2x(x+3) + (x+3)

Step 5: If we've done this correctly, our two new terms should have a clearly visible common factor.

In this case we can see that (x+3) is common to both terms, so we can go:

Start with:2x(x+3) + (x+3)

Which is:2x(x+3) + 1(x+3)

And so:(2x+1)(x+3)

Step 6 : Equate each factor to zero

(2x + 1) = 0

2x = -1

x = -1/2, and

(x+3) = 0

x = -3

So Roots of quadratic equation 2x²+7x+3 are x = -1/2 or x = -3.

Solving Quadratic Equation by Completing the Square

In the method of completing the squares, the quadratic equation is expressed in the form

(x $\pmk)$ ² =p²

Example : Solve quadratic equation 2x²+ 24x = -64 by completing the square .

Solution :

(i) Express the quadratic equation in standard form.

2x²+ 24x +64 =0

(ii) Divide the equation by the coefficient of x² to make the coefficient of x² equal to 1

divide the equation with 2

x²+ 12x +32=0

x²+ 12x = -32

(iii) Add the square of half of the coefficient of x to both sides of the equation to get an expression of the form x² $\pm 2kx +$ $k$ ² and obtain the equation of the form(x $\pmk)$ ² =p² .

x²+ 12x + 36 = 36 - 32

x²+ 12x +36= 4

(x+6)(x+6)=4

(x+6)² = 2²

(iv) Take the positive and negative square roots.

(x+6) $\pm 2$

$solving forward$

x+6=2

x= 2-6 = -4

and , x+6 = -2

x= -6-2 = -8

Thus , x = -4 and -8

Solving Quadratic Equation Using Quadratic Formula

Quadratic Formula is used to directly obtain the roots of a quadratic equation from the standard form of the equation.

For the quadratic equation $a x^{2} + b x + c = 0,$

where a, b and c are the real numbers and b²– 4ac is called discriminant.

To find the roots of the equation, put the value of a, b and c in the quadratic formula.

By substituting the values of a,b and c, we can directly get the roots of the equation.

Discriminant

For a quadratic equation of the form $a x^{2} + b x + c = 0,$ the expression $b^{2} - 4 a c$ is called the discriminant, (denoted by D), of the quadratic equation.

The discriminant determines the nature of roots of the quadratic equation based on the coefficients of the quadratic equation.

Nature of graph for different values of Discriminant

Based on the value of the discriminant, $D = b^{2} - 4 a c$ , the roots of a quadratic equation can be of three types.

Case 1: If D>0, the equation has two distinct real roots.

If D>0, the parabola cuts the x-axis at exactly two distinct points.
The roots are distinct.
This case is shown in the below figure where the quadratic polynomial cuts the x-axis at two distinct points.

Example : Solve the Quadratic Equation x

^{2}

- 6x -16 = 0

Solution :

Case 2: If D=0, the equation has two equal real roots.

If D=0, the parabola just touches the x-axis at one point and the rest of the parabola lies above or below the x-axis.
In this case, the roots are equal.

This case is shown in the below figure where the quadratic polynomial touches the x-axis at only one point.

Example : Solve the Quadratic Equation -4x $^{2}$ + 12x -9 = 0

Solution :

Case 3: If D<0, the equation has no real roots.

If D<0, the parabola lies entirely above or below the x-axis and there is no point of contact with the x-axis.
In this case, there are no real roots.

This case is shown in the below figure where the quadratic polynomial neither cuts nor touches the x-axis.

Example : Solve the Quadratic Equation x $^{2}$ + 25 = 0

Solution :

From the Graphical Representation of a Quadratic Equation as shown in the above examples, we can conclude that ,

The graph of a quadratic polynomial is a parabola.
The roots of a quadratic equation are the points where the parabola cuts the x-axis i.e. the points where the value of the quadratic polynomial is zero.

For a quadratic polynomial $a x^{2} + b x + c,$

If a>0, the parabola opens upwards.
If a<0, the parabola opens downwards.
If a = 0, the polynomial will become a first-degree polynomial and its graph is linear.

Formation of a quadratic equation from its roots

To find out the standard form of a quadratic equation when the roots are given:
Let $α$ and $β$ be the roots of the quadratic equation $a x^{2} + b x + c = 0$ . Then,

$(x - α) (x - β) = 0$

On expanding, we get,

$x^{2} - (α + β) x + α β = 0,$

which is the standard form of the quadratic equation.

Here, $a = 1, b = - (α + β) a n d c = α β .$

Sum and Product of Roots of a Quadratic equation

Let $α a n d β$ be the roots of the quadratic equation $a x^{2} + b x + c = 0.$ Then,

Sum of roots $= α + β$ =-b/a

Product of roots $= α β$ = c/a

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