2. Quadratic Equations
Introduction to Polynomials
Introduction to Quadratic Equations
Forming Quadratic Equations
Roots of a Quadratic Equation
Solving Quadratic Inequalities
Sum and Product of the roots
Discriminant and the nature of roots
Maximum and minimum value of a
Quadratic expression
Common roots
Higher Order Equations
Coefficients and roots
Descartes rule of signs
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Quadratic Equation 2
4. Polynomials
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π(π₯) = π1 π₯ π + π2 π₯ πβ1
+ β― . πππ₯ is a polynomial in
x,
If π1, π2, π3β¦. are real numbers,
π₯ is a real variable and
π is a whole number
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5. Classification of Polynomials
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A. On the basis of coefficients
I. Polynomials over integers
Eg: 3x2 + 4x + 9
II. Polynomials over rational numbers
Eg: 3/5 x2 + 4x + 9
III. Polynomial over real numbers
Eg: β3 x2 + β7 x +9
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6. Classification of Polynomials
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B. On the basis of number of terms
I. Monomials
Eg: 3x2 , β7 x
II. Binomials
Eg: 3/5 x2 + 4x
III. Trinomials
Eg: β3 x2 + β7 x +9
IV. Polynomials
Usually more than three terms
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C. On the basis of the degree
I. Linear polynomial
Eg: 4x+ 3y
II. Quadratic Polynomial
Eg: 4x2 +8xy
III. Cubic polynomial
Eg: 8 x3 + β7 x +9
III. Biquadratic polynomial
Eg: 8 x4 + β7 x3 +9
7Classification of Polynomials
12. Forming a quadratic equation
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1.When roots are given
2.When sum of the roots and the product of the roots are
given
3.When the roots are related to the roots of another
quadratic equation
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13. Forming a quadratic equation
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1. When roots are given
Form a quadratic equation whose roots are 1 and 2
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Ans: π₯2
+ 3π₯ + 2 =
0
14. Forming a quadratic equation
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2. When sum of the roots and the product of the roots are given
Form a quadratic equation such that the sum of the roots is 4 and the
product of the roots is 3
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Ans: π₯2
β 4π₯ + 3 =
0
15. Forming a quadratic equation
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3. When the roots are related to the roots of another quadratic equation
Form a quadratic equation whose roots are two more than the roots of the equation x2 -
3x +2 = 0
Changed roots Changed Q.Eq.
1 Ξ± + p and Ξ² + p a (x - p)2 + b (x - p) + c =0
2 Ξ± - p and Ξ² - p a (x + p)2 + b (x + p) + c =0
3 Ξ±p and Ξ²p a (x / p)2 + b (x / p) + c =0
4 Ξ±/p and Ξ²/p a (x p)2 + b (x p) + c =0
5 1/ Ξ± and 1/ Ξ² a (1/x )2 + b (1/x ) + c =0
6 -Ξ± and -Ξ²
a (-x )2 + b (-x ) + c =0
a x 2 - b x + c =0
7 Ξ±2 and Ξ²2 ax + b root x + c =0
8 Ξ±n and Ξ²n ax2/n + bx1/n + c=0
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16. β’ Roots of a Quadratic Equation
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17. What is a root of a quadratic equation?
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18. Finding roots of a quadratic equation
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1. Splitting the middle term
2. Quadratic formula
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19. Finding roots of a quadratic equation
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1. Find the roots of the quadratic equation x2 + 5x + 6 = 0
2. Find the roots of the quadratic equation 6x2 - 5x - 6 = 0
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Answers:
1. -2 or -3
2. 3/2 0r -2/3
20. Finding roots of a quadratic equation
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ο Find the roots of the quadratic equation x2 + 6x + 10 = 0
ο Quadratic Formula
ο Roots =
βπ Β± π2β4ππ
2π
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Solve for x in x2 β 5x +6 >0
Solve for x in x2 β 5x +6 <0
Solve for x in x2 β 5x +6 β₯0
Solve for x in x2 β 5x +6 β€0
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Solve for x in
π₯2 β 5π₯ + 6
(π₯ + 2)(4π₯ β 1)
> 0
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Answer:
(ββ, β2) βͺ β1, ΒΌ βͺ (6, β)
24. β’ Sum and product of roots of a Quadratic Equation
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Problems
Find the value of p if one root of the quadratic equation
π₯2 β 12π₯ β π = 0 is twice the other
1) 32 2) -32 3) 390 4) 450
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Answer: -32
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Problems
Aakash and Kiran noted down a quadratic equation from the
blackboard. Aakash made an error while noting down the
coefficient of x and got the roots as 12 and 4, Kiran made
an error in noting down the constant term and got the roots
as 9 and 5. Which of the following is the actual equation?
A. π₯2 β 14π₯ + 14 = 0 B.2π₯2 + 14π₯ β 24 = 0
C. π₯2 β 14π₯ + 48 = 0 D. 3π₯2 β 17π₯ + 48 = 0
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Answer: Option C
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Discriminant and nature of roots of a Q. Eq
If the coefficients are rational, irrational roots occur in pairs.
If the coefficients are real, complex roots occur in pairs.
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Problems
If the roots of 8 π π₯2 + 8π₯ + 32 π = 0 are real and
equal, find the value of m
1. 2
2. Β½
3. ΒΎ
4. None of these
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Answer: 1/2
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Problems
How many equations of the form π₯2 + 8π₯ + π exist such
that the roots are real and π is a positive integer?
1. 15
2. 16
3. 17
4. None of these
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Answer: 16
36. β’ Maximum and minimum value of a
Quadratic expression
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Maximum Value or Minimum Value
A quadratic function Ζ(x) attains a maximum of 3 at x = 1.
The value of the function at x = 0 is 1.
What is the value Ζ(x) at x = 10?
1. -119
2. -159
3. -110
4. -180
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Answer : option 2
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Relationship between coefficients and roots
Consider the cubic equation
ππ₯3 + ππ₯2 + ππ₯ + π = 0
β’ Sum of roots = β π/π
β’ Sum of all the three pairs of two roots taken at a time =
c/a
β’ Product of all the roots = - d/a
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Relationship between coefficients and roots
In general for any equation of degree n
β’ Sum of roots = (-1)coefficient of x(n-1)/coefficient of xn
β’ Sum of all the pairs of roots taken 2 at a time =
coefficient of x(n-2)/coefficient of xn
β’ Sum of roots taken r at a time =
(-1)r coefficient of x(n-r)/coefficient of xn
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48. Number of positive and negative real roots in a polynomial
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49. Problems
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ο Use Descartes' Rule of Signs to determine the
number of real roots of
π (π₯) = π₯5 β π₯4 + 3π₯3 + 9π₯2 β π₯ + 5
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Answer: 4 or 2 or 0 positive real roots and 1 negative real root
Editor's Notes
Substituting 2+ 3 in the equation and substituting the value of q, we get the value of p as 6+ 3 . Hence the other root is 4
