## Quadratic Equations

An equation in which the highest power of the unknown quantity is two is called **quadratic equation**.

### Types of quadratic equation

Quadratic equations are of two types:

Purely quadratic | Adfected quadratic |

ax^{2} + c = 0 where a, c ∈ C and b = 0, a ≠ 0 | ax^{2} + bx + c = 0 where a, b, c ∈ C and a ≠ 0, b ≠ 0 |

**Roots of a quadratic equation:**

The values of variable x which satisfy the quadratic equation is called roots of quadratic equation.

### Solution of quadratic equation

**(1) Factorization method**

Let ax^{2} + bx + c = a(x – α)(x – β) = 0.

Then x = α and x = β will satisfy the given equation.

Hence, factorize the equation and equating each factor to zero gives roots of the equation.

**Example:** 3x^{2} – 2x + 1 = 0 ⇒(x – 1)(3x + 1) = 0; x = 1, -1/3

**(2) Sri Dharacharya method **

By completing the perfect square as

Every quadratic equation has two and only two roots.

### Nature of roots

In a quadratic equation ax^{2} + bx + c = 0, let us suppose that are real and a ≠ 0. The following is true about the nature of its roots.

- The equation has real and distinct roots if and only if D ≡ b
^{2}– 4ac > 0. - The equation has real and coincident (equal) roots if and only if D ≡ b
^{2}– 4ac = 0. - The equation has complex roots of the form α ± iβ, α ≠ 0 if and only if D ≡ b
^{2}– 4ac < 0. - The equation has rational roots if and only if a, b, c ∈ Q (the set of rational numbers) and D ≡ b
^{2}– 4ac is a perfect square (of a rational number). - The equation has (unequal) irrational (surd form) roots if and only if D ≡ b
^{2}– 4ac > 0 and not a perfect square even if a, b and c are rational. In this case if p + √q, p,q rational is an irrational root, then p – √q is also a root (a, b, c being rational). - α + iβ (β ≠ 0 and α, β ∈ R) is a root if and only if its conjugate α – iβ is a root, that is complex roots occur in pairs in a quadratic equation. In case the equation is satisfied by more than two complex numbers, then it reduces to an identity.

0.x^{2}+ 0.x + 0 = 0, i.e., a = 0 = b = c.

### Relations between roots and coefficients

(1) Relation between roots and coefficients of quadratic equation: If α and β are the roots of quadratic equation , (a ≠ 0) then

(2) Formation of an equation with given roots: A quadratic equation whose roots are α and β is given by (x – α)(x – β) = 0.

∴ x^{2} – (α+iβ)x + αβ = 0

i.e. x^{2} – (sum of roots)x + (product of roots) = 0

∴ x^{2} – Sx + P = 0

(3) Symmetric function of the roots : A function of α and β is said to be a symmetric function, if it remains unchanged when α and β are interchanged.

For example, α^{2} + β^{2} + 2αβ is a symmetric function of α and β whereas α^{2} + β^{2} + 2αβ is not a symmetric function of α and β.

In order to find the value of a symmetric function of α and β, express the given function in terms of α + β and αβ. The following results may be useful.

### Properties of quadratic equation

- If f(a) and f(b) are of opposite signs then at least one or in general odd number of roots of the equation lie between a and b.
- If then f(a) = f(b) there exists a point c between a and b such that f(c) = 0, a<c<b.
- If α is a root of the equation f(x) = 0 then the polynomial f(x) is exactly divisible by (x – α), then (x – α) is factor of f(x).
- If the roots of the quadratic equations a
_{1}x^{2}+ b_{1}x + c_{1}= 0 and a_{2}x^{2}+ b_{2}x + c_{2}= 0 are in the same ratio [i.e. α_{1}/β_{1}= α_{2}/β_{2}] then b_{1}^{2}/b_{2}^{2}= a_{1}c_{1}/a_{2}c_{2}.