MATHS PROJECT WORK MADE BY-MAYANK GOEL Quadratic Equation

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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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But what about NASTY trinomials that don’t factor? Abu Ja'far Muhammad ibn Musa Al-Khwarizmi Born : about 780 in Baghdad (Iraq) Died : about 850 The Challenge

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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 :

By quadratic equation:

By quadratic equation

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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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