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two distinct real roots, if b²-4ac>0, two equal roots , if b²-4ac=0, and no real roots, if b²-4ac<0. A quadratic equation ax²+bx+c=0 has

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The Quadratic Equations If p(x) is a quadratic polynomial, then p(x)=0 is called a quad-ratic equation. A polynomial with degree 2 is called a quadratic polynomial. example 2x ²+4x+7

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A real no. α is said to be a root of a quadratic equations ax²+bx+c=0 , if a α ²+b α +c=0. The zeroes of a quadratic polynomial ax ²+bx+c and the roots of the quadratic equation ax ²+bx+c=0 are the same . A quadratic equation in the variable x is of the form ax ²+bx+c=0,wherea,b,c are real numbers and a≠0. INTRODUCTION

### Identifying Terms:

Example f(x)=5x 2 -7x+1 Quadratic term 5x 2 Linear term -7x Constant term 1 Identifying Terms

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If α and β(p and q, x 1 and x 2 ) are the roots of ax 2 + bx + c = 0, then sum of roots = α + β and product of roots = αβ

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The ROOTS (or solutions ) of a polynomial are its x-intercepts Recall: The x-intercepts occur where y = 0 . What it means FURTHER

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FACTORIZATION METHOD COMPLETING THE SQUARE QUADRATIC FORMULA (SHREEDHARACHARYA’S RULE) 3 method of solving

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SOLVING BY FACTORIZATION METHOD

### FACTORIZATION METHOD:

FACTORIZATION METHOD Example: Find the roots: y = x ² + x – 6 = 0 Solution: Factoring: y = (x + 3)(x - 2) 0 = (x + 3)(x - 2) The roots are: x = -3; x = 2

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SOLVING BY COMPLETING THE SQUARE

### COMPLETING THE SQUARE :

COMPLETING THE SQUARE we have , 9x²-15x+6=0 or, x²-15/9x+6/9=0 or, x²-5/3x+2/3=0 or, x²-5/3x=-2/3 or, x²-2(5/6)x+(5/6)² =(5/6)²-2/3 or, (x-5/6)²=25/36-2/3 or, ( x-5/6)²=25-24/36 or, (x-5/6)²=1/36 or, x-5/6=±1/6 or, x=5/6±1/6 or, x=5/6+1/6=1 or, 5/6-1/6=4/6=2/3 x=1 or, x=2/3 Example: Find the roots: 9x²-15x+6=0 Solution: Hence, the roots of the equation are 1 and 2/3

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After centuries of work, mathematicians realized that as long as you know the coefficients, you can find the roots of the quadratic. Even if it doesn’t factor! The Formula

### What Does The Formula Do ?:

What Does The Formula Do ? The Quadratic formula allows you to find the roots of a quadratic equation (if they exist) even if the quadratic equation does not factorise. The formula states that for a quadratic equation of the form : ax 2 + bx + c = 0 The roots of the quadratic equation are given by :

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△ = b 2 - 4 ac Since the expression b 2 - 4 ac can be used to determine the nature of the roots of a quadratic equation in the form ax 2 – bx + c = 0, it is called the discriminant of the quadratic equation.

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a = b = c = 1 10 -7

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Another Example Use the quadratic formula to solve the equation : x 2 + 5x + 6= 0 Solution: x 2 + 5x + 6= 0 a = 1 b = 5 c = 6 x = - 2 or x = - 3 These are the roots of the equation.

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Thank you 