A polynomial of degree 2 (i.e.ax^{2} + bx + c) is called a **quadratic polynomial** where a ≠ 0 and a, b, c are real numbers.

The general form of quadratic equation is ax^{2} + bx + c = 0, where a,b,c are real numbers and a ≠ 0 Since a ≠ 0, quadratic equations, in general are of the following types :-

(i) b = 0, c ≠ 0i.e., of the type ax^{2} + c = 0.

(ii) b ≠ 0c = 0, i.e. of the type ax^{2} + bx = 0.

(iii) b = 0, c = 0, i.e. of the type ax^{2} = 0.

(iv) b ≠ 0, c ≠ 0i.e., of the type ax^{2 }+ bx + c = 0.

**It is classified into two categories:**

(i) Pure quadratic equation (of the form ax^{2} + c = 0 i.e., b = 0 in ax^{2} + bx + c)

e.g. x^{2 }- 4 = 0 and 3x^{2} + 1 = 0 are pure quadratic equations.

(ii) Affected quadratic equation (of the form ax^{2} + bx + c = 0, b ≠ 0).

e.g. x^{2}- 2x - 8 = 0 and 5x^{2} + 3x - 2 = 0 are affected quadratic equations.

For a quadratic polynomial p(x) = ax^{2} + bx + c, those values of x for which ax^{2} + bx + c = 0 is satisfied, are called zeros of quadratic polynomial p(x), i.e. if p(α) = aα^{2} + bα + c = 0 then α is called the zero of quadratic polynomial.

The value of x which satisfies the given quadratic equation is known as its root. The roots of the given equation are known as its solution.

**General form of a quadratic equation is :**

ax^{2} + bx + c = 0

or 4a^{2}x^{2} + 4abx + 4ac = 0 [Multiplying by 4a]

or 4a^{2}x^{2} + 4abx = –4ac [By adding b^{2} both sides]

or 4a^{2}x^{2} + 4abc + b^{2} = b^{2} – 4ac

or (2ax + b)^{2} = b^{2} – 4ac

Taking square root of both the sides

2ax + b =

or

Hence, roots of the quadratic equation ax2 + bx + c = 0 are

A quadratic equation is satisfied by exactly two values of ‘a’ which may be real or imaginary. The equation, ax2 + bx + c = 0 is :

- A quadratic equation if a ≠ 0 Two roots
- A linear equation if a = 0, b ≠ 0 One root
- A contradiction if a = b = 0, c ≠ 0 No root
- An identify if a = b = c = 0 Infinite roots
- A quadratic equation cannot have more than two roots.
- If follows from the above statement that if a quadratic equation is satisfied by more than two values of x, then it is satisfied by every value of x and so it is an identity.

Consider the quadratic equation, ax^{2} + bx + c = 0 having αβ as its roots and b^{2} – 4ac is called discriminate of roots of quadratic equation. It is denoted by D or Δ.

Roots of the given quadratic equation may be

(i) Real and unequal (ii) Real and equal (iii) Imaginary and unequal.

Let the roots of the quadratic equation ax2 + bx + c = 0 (where a ≠ 0,b,c ∈ R ) be α and β then

The nature of roots depends upon the value of expression ‘b2 - 4ac’ within the square root sign. This is known as discriminate of the given quadratic equation.

**Consider the Following Cases:**

Case-1 When b^{2 }- 4ac > 0, (D > 0)

In this case roots of the given equation are real and distinct and are as follows

(i) When a ≠ 0,b,c ∈ Q and b^{2} - 4ac is a perfect square

In this case both the roots are rational and distinct.

(ii) When a ≠ 0,b,c ∈ Q and b^{2 }- 4ac is not a perfect square

In this case both the roots are irrational and distinct. [See remarks also]

Case-2 When b^{2} - 4ac = 0, (D = 0)

In this case both the roots are real and equal to -b/2a.

Case-3 When b^{2 }- 4ac < 0, (D < 0)

In this case b^{2} – 4ac < 0, then 4ac – b^{2} > 0

i.e. in this case both the root are imaginary and distinct.

- If a,b,c ∈ Q and b2 - 4ac is positive (D > 0) but not a perfect square, then the roots are irrational and they always occur in conjugate pairs like 2 + √3 and 2- √3. However, if a, b, c are irrational number and b2 – 4ac is positive but not a perfect square, then the roots may not occur in conjugate pairs.
- If 2 – 4ac is negative (D > 0), then the roots are complex conjugate of each other. In fact, complex roots of an equation with real coefficients always occur in conjugate pairs like 2 + 3i and 2 – 3i. However, this may not be true in case of equations with complex coefficients. For example, x2 – 2ix – 1 = 0 has both roots equal to i.
- If a and c are of the same sign and b has a sign opposite to that of a as well as c, then both the roots are positive, the sum as well as the product of roots is positive (D ≥ 0).
- If a,b, are of the same sign then both the roots are negative, the sum of the roots is negative but the product of roots is positive (D ≥ 0).

**question **1. Check whether the following equations are quadratic or not.

(i) (ii)

(iii) (iv)

**Solution: ** (i) can be rewritten as

It is of the form ax^{2} + bx + c = 0

Therefore, the given equation is a quadratic equation.

(ii) As and (x + 2) (x - 2) = x^{2} - 4

Therefore, we have x^{2} + x + 8 = x^{2} - 4

⇒ x + 12 = 0

It is not of the form ax^{2} + bx + c = 0

Therefore, the given equation is not a quadratic equation.

(iv)

⇒ x + x^{-1} - 2 = 0

Since x + x^{-1} - 2 is not a quadratic polynomial.

∴ is not a quadratic equation

**question **2. In each of the following, determine the value of k for which the given value is a solution of the equation.

(i)

(ii)

**Solution: ** (i) …(i)

Since x = 2 is a solution of the equation

Now substituting x = 2 in equation (i), we get

(ii) Since x = –2 is a solution of the equation

…(i)

Now substituting x = – 2 in (i), we get

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