Maths Class 10 Notes for Quadratic Equations

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) = ax2 + 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 ax2 + bx + c 0 where a ≠ 0, b ≠ 0,c ≠ 0
(ii) Pure quadratic equation : An equation in the form of ax2 = 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 x2 — 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 ax2 + bx + c =0, a 0 if aα2 + 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 ax2 + bx + c and the roots of the quadratic equation ax2 + bx + c = 0 are the same.
2. Roots of quadratic equation ax2 + 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 x2 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 — b2 = 0 and then we can easily find its roots.

DISCRIMINANT

The expression b2 — 4ac is called the discriminant of the quadratic equation.

SOLUTION OF A QUADRATIC EQUATION BY DISCRIMINANT METHOD

Let quadratic equation is ax2 + bx + c = 0

Step 1. Find D = b2 — 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 ax2 + bx + c = 0 (a ≠ 0).

Thus, if b2 — 4ac ≥ 0, then the roots of the quadratic

—b ± √b2 — 4ac / 2a equation are given by

QUADRATIC FORMULA

—b ± √b2 — 4ac / 2a is known as the quadratic formula

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

NATURE OF ROOTS

(i) If b2 — 4ac > 0, then the roots are real and distinct.

(ii) If b2 — 4ac = 0, the roots are real and equal or coincident.

(iii) If b2 — 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 x2 — (α + β)x + αβ =0

NOTE :

Let α and β be two roots of the quadratic equation (ax2 + bx + c = 0 then

Sum of Roots: – the coefficient of x / the coefficient t of x2 ⇒ α + β = – b / a

Product of Roots :

αβ = constant term / the coefficient t of x2 ⇒ αβ = c / a

METHOD OF SOLVING WORD PROBLEMS

Step 1: Translating the word problem into Mathematics form (symbolic form) according to the given condition

Step 2 : Form the word problem into Quadratic equations and solve them.

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