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Expression, Zeroes, Roots, Formation, Example, Solution | Algebra - Quadratic Equations | 10th Mathematics : Algebra

Chapter: 10th Mathematics : Algebra

Quadratic Equations

1. Zeroes of a Quadratic Polynomial 2. Roots of Quadratic Equations 3. Formation of a Quadratic Equation

Quadratic Equations

Introduction

Arab mathematician Abraham bar Hiyya Ha-Nasi, often known by the Latin name Savasorda, is famed for his book ‘Liber Embadorum’ published in 1145 AD(CE) which is the first book published in Europe to give the complete solution of a quadratic equation.

For a period of more than three thousand years beginning from early civilizations to current times, humanity knew how to solve a general quadratic equation in terms of its co-efficients by using four arithmetical operations and extraction of roots. This process is called “Solving by Radicals”. Huge amount of research has been carried to this day in solving various types of equations.

Quadratic Expression

An expression of degree n in variable x is a0xn+ a1xn−1+a2xn−2+ ... +an-1x +an where a0 ≠0  and  a1 , a2,...an are real numbers. a 0 , a1, a 2 , ... an are called coefficients of the expression.

In particular an expression of degree 2 is called a Quadratic Expression which is expressed as p(x) = ax 2 +bx +c, a ≠0 and a, b, c are real numbers.

 

1. Zeroes of a Quadratic Polynomial

Let p(x) be a polynomial. x=a is called zero of p(x) if p(a) = 0

For example, if p(x) = x2–2x–8 then p(–2)=4+4–8 = 0 and p(4) = 16– 8 –8 = 0

Therefore –2 and 4 are zeros of the polynomial p(x) = x2–2x–8.

 

2. Roots of Quadratic Equations

Let ax2 + bx +c = 0, (a ≠ 0) be a quadratic equation. The values of x such that the expression ax 2 + bx + c becomes zero are called roots of the quadratic equation ax 2 + bx +c = 0 .

We have, ax 2 + bx + c = 0


 

3. Formation of a Quadratic Equation

If a and b are roots of a quadratic equation ax2 + bx +c = 0 then


Since, (x - a) and (x - b) are factors of ax2 + bx +c = 0 ,

We have (x - α)(x - β) = 0

Hence, x 2 (α + β)x + αβ = 0

That is, x2 - (sum of roots) x  + product of roots = 0 is the general form of the quadratic equation when the roots are given.

 

Example 3.24

Find the zeroes of the quadratic expression x2 + 8x + 12.

Solution

Let p(x)= x2 + 8x + 12 =(x+2)(x+6)

p(–2)= 4 – 16 + 12=0

p(–6)= 36 – 48 + 12=0

Therefore –2 and –6 are zeros of p(x)= x2 + 8x + 12

 

Example 3.25

Write down the quadratic equation in general form for which sum and product of the roots are given below.

(i) 9, 14

(ii) – 7/2 , 5/2

(iii) – 3/5 , - 1/2

Solution

(i) General form of the quadratic equation when the roots are given is x2 - (sum of the roots ) x + product of the roots = 0

x 2 9x + 14 = 0


Therefore, 10x 2 + 6x − 5 = 0.

 

Example 3.26

Find the sum and product of the roots for each of the following quadratic equations :

(i) x 2 + 8x − 65 = 0

(ii) 2x2 + 5x + 7 = 0

(iii) kx 2 k 2 x 2k 3  = 0

Solution 

Let a and b be the roots of the given quadratic equation

x 2 + 8x 65 = 0

a = 1,  b = 8,  c = –65

α + β = −b/a = –8 and αβ = c/a  = –65

α + β = −8 αβ = −65

(ii) 2x 2 + 5x + 7 = 0

a = 2,  b = 5,  c = 7


(iii) kx 2k 2 x − 2k 3  = 0

a = k,  b = -k2 ,  c = –2k3


 

Tags : Expression, Zeroes, Roots, Formation, Example, Solution | Algebra , 10th Mathematics : Algebra
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10th Mathematics : Algebra : Quadratic Equations | Expression, Zeroes, Roots, Formation, Example, Solution | Algebra


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