**Introduction to Quadratic Equations**

**Quadratic Polynomial**

A polynomial of the form ax2+bx+c, where a,b and c are real numbers and a≠0 is called a quadratic polynomial.

**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 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 ax2+bx+c=0, then aα2+bα+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 x2−4=0:

**Graph of a Quadratic Equation**

In the above figure, -2 and 2 are the roots of the quadratic equation x2−4=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 2×2−5x+3=0

⇒2×2−2x−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 x2 and the constant.

(-2) + (-3) = (-5)

And (-2) × (-3) = 6

2×2−2x−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.

**Solving QE by Completing the Square**

**Solving a Quadratic Equation by Completion of squares method**

In the method of completing the squares, the quadratic equation is expressed in the form (x±k)2=p2.

Consider the quadratic equation 2×2−8x=10

(i) Express the quadratic equation in standard form.

2×2−8x−10=0

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

x2−4x−5=0

(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±2kx+k2.

(x2−4x+4)−5=0+4

(iv) Isolate the above expression, (x±k)2 on the LHS to obtain an equation of the form (x±k)2=p2

(x−2)2=9

(v) Take the positive and negative square roots.

x−2=±3

x=−1 or x=5

**Solving QE Using Quadratic Formula**

**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 ax2+bx+c=0,

x= [-b± √(b2-4ac)]/2a

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 ax2+bx+c=0, the expression b2−4ac 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.

**Solving using Quadratic Formula when D>0**

Solve 2×2−7x+3=0 using the quadratic formula.

(i) Identify the coefficients of the quadratic equation. a = 2,b = -7,c = 3

(ii) Calculate the discriminant, b2−4ac

D=(−7)2−4×2×3= 49-24 = 25

D> 0, therefore, the roots are distinct.

(iii) Substitute the coefficients in the quadratic formula to find the roots

x= [-(-7)± √((-7)2-4(2)(3))]/2(2)

x=(7 ±5)/4

x=3 and x= 1/2 are the roots.

**Nature of Roots**

Based on the value of the discriminant, D=b2−4ac, the roots of a quadratic equation can be of three types.

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

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

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

**Graphical Representation of a Quadratic Equation**

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.

Now, the graph of x2+5x+6=0 is:

In the above figure, -2 and -3 are the roots of the quadratic equation

x2+5x+6=0.

For a quadratic polynomial ax2+bx+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.

The discriminant, D=b2−4ac

**Nature of graph for different values of D.**

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

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 above figure in b, where the quadratic polynomial touches the x-axis at** only one point**.

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 above figure in c, where the quadratic polynomial **neither cuts nor touches **the x-axis.

**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 ax2+bx+c=0. Then,

(x−α)(x−β)=0

On expanding, we get,

x2−(α+β)x+αβ=0, which is the standard form of the quadratic equation. Here, a=1,b=−(α+β) and c=αβ.

**Sum and Product of Roots of a Quadratic equation**

Let α and β be the roots of the quadratic equation ax2+bx+c=0. Then,

Sum of roots =α+β=-b/a

Product of roots =αβ= c/a