- Books Name
- Mathematics Book for CBSE Class 10

- Publication
- Carrier Point

- Course
- CBSE Class 10

- Subject
- Mathmatics

**QUADRATIC EQUATIONS**

The polynomial of degree two is called quadratic polynomial and equation corresponding to a quadratic polynomial P(x) is called a quadratic equation in variable x.

Thus, P(x) = ax^{2} + bx + c =0, a ≠ 0, a, b, c ∈ R is known as the standard form of quadratic equation.

There are two types of quadratic equation.

(i)** Complete quadratic equation :** The equation ax^{2} + bx + c 0 where a ≠ 0, b ≠ 0,c ≠ 0

(ii) **Pure quadratic equation :** An equation in the form of ax^{2} = 0, a ≠ 0, b = 0, c = 0

**ZERO OF A QUADRATIC POLYNOMIAL**

The value of x for which the polynomial becomes zero is called zero of a polynomial

For instance,

1 is zero of the polynomial x^{2} — 2x + 1 because it become zero at x = 1.

**SOLUTION OF A QUADRATIC EQUATION BY**

**FACTORISATION**

A real number x is called a root of the quadratic equation ax^{2} + bx + c =0, a 0 if aα^{2} + bα + c =0.In this case, we say x = α is a solution of the quadratic equation.

**NOTE:**

1. The zeroes of the quadratic polynomial ax^{2} + bx + c and the roots of the quadratic equation ax^{2} + bx + c = 0 are the same.

2. Roots of quadratic equation ax^{2} + bx + c =0 can be found by factorizing it into two linear factors and equating each factor to zero.

**SOLUTION OF A QUADRATIC EQUATION BY COMPLETING THE SQUARE**

By adding and subtracting a suitable constant, we club the x^{2} and x terms in the quadratic equation so that they become complete square, and solve for x.

In fact, we can convert any quadratic equation to the form (x + a)^{2} — b^{2} = 0 and then we can easily find its roots.

**DISCRIMINANT**

The expression b^{2} — 4ac is called the discriminant of the quadratic equation.

**SOLUTION OF A QUADRATIC EQUATION BY DISCRIMINANT METHOD**

Let quadratic equation is ax^{2} + bx + c = 0

**Step 1.** Find D = b^{2} — 4ac.

**Step 2.**

(i) If D > 0, roots are given by

x = -b + √D / 2a , -b – √D / 2a

(ii) If D = 0 equation has equal roots and root is given by x = -b / 2a.

(iii) If D < 0, equation has no real roots.

**ROOTS OF THE QUADRATIC EQUATION**

Let the quadratic equation be ax^{2} + bx + c = 0 (a ≠ 0).

Thus, if b^{2} — 4ac ≥ 0, then the roots of the quadratic

—b ± √b^{2} — 4ac / 2a equation are given by

**QUADRATIC FORMULA**

—b ± √b^{2} — 4ac / 2a is known as the quadratic formula

which is useful for finding the roots of a quadratic equation.

**NATURE OF ROOTS**

(i) If b^{2} — 4ac > 0, then the roots are** real and distinct.**

(ii) If b^{2} — 4ac = 0, the roots are** real and equal or coincident.**

(iii) If b^{2} — 4ac <0, the roots are not** real (imaginary roots)**

**FORMATION OF QUADRATIC EQUATION WHEN TWO ROOTS ARE GIVEN**

If α and β are two roots of equation then the required quadratic equation can be formed as x^{2} — (α + β)x + αβ =0

**NOTE :**

Let α and β be two roots of the quadratic equation (ax^{2} + bx + c = 0 then

**Sum of Roots:** – the coefficient of x / the coefficient t of x^{2} ⇒ α + β = – b / a

**Product of Roots :**

αβ = constant term / the coefficient t of x^{2} ⇒ αβ = c / a