The document discusses the quadratic equation and the relationship between its roots and coefficients. It defines the quadratic equation as a second degree polynomial equation. The roots of a quadratic equation, also called alpha and beta, are related to the coefficients in the following ways:
1) The sum of the roots is equal to the negative of the coefficient of the second term divided by the leading coefficient.
2) The product of the roots is equal to the constant term divided by the leading coefficient.
3) Examples are provided to demonstrate calculating the sum and product of roots given a quadratic equation.
Content:
1- Mathematical proof (what and why)
2- Logic, basic operators
3- Using simple operators to construct any operator
4- Logical equivalence, DeMorgan’s law
5- Conditional statement (if, if and only if)
6- Arguments
Basic Concept of discrete math. I discuss a few topic of basic discrete math. I think you get a clear concept. Topics are definition,story of discrete math, Importance of discrete math, and many other basic topic.
Content:
1- Mathematical proof (what and why)
2- Logic, basic operators
3- Using simple operators to construct any operator
4- Logical equivalence, DeMorgan’s law
5- Conditional statement (if, if and only if)
6- Arguments
Basic Concept of discrete math. I discuss a few topic of basic discrete math. I think you get a clear concept. Topics are definition,story of discrete math, Importance of discrete math, and many other basic topic.
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dug84356_ch09a.qxd 9/14/10 2:11 PM Page 557
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9.1
9.2
9.3
9.4
9.5
9.6
Radicals
Rational Exponents
Adding, Subtracting, and
Multiplying Radicals
Quotients, Powers,
and Rationalizing
Denominators
Solving Equations with
Radicals and Exponents
Complex Numbers
9
Radicals and Rational
Exponents
Just how cold is it in Fargo, North Dakota, in winter? According to local meteorol
ogists, the mercury hit a low of –33°F on January 18, 1994. But air temperature
alone is not always a reliable indicator of how cold you feel. On the same date,
the average wind velocity was 13.8 miles per hour. This dramatically affected how
cold people felt when they stepped outside. High winds along with cold temper
atures make exposed skin feel colder because the wind significantly speeds up
the loss of body heat. Meteorologists use the terms “wind chill factor,”“wind chill
index,” and “wind chill temperature” to take into account both air temperature
and wind velocity.
Through experimentation in Ant
arctica, Paul A. Siple developed a
formula in the 1940s that measures the
wind chill from the velocity of the wind
and the air temperature. His complex
formula involving the square root of
the velocity of the wind is still used
today to calculate wind chill temper
atures. Siple’s formula is unlike most
scientific formulas in that it is not
based on theory. Siple experimented
with various formulas involving wind
velocity and temperature until he
found a formula that seemed to predict
how cold the air felt.
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Siple s formula is stated
and used in Exercises 111
and 112 of Section 9.1.
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558 Chapter 9 Radicals and Rational Exponents 9-2
9.1 Radicals
In Section 4.1, you learned the basic facts about powers. In this section, you will
study roots and see how powers and roots are related.
In This Section
U1V Roots
U2V Roots and Variables
U3V Product Rule for Radicals
U4V Quotient Rule for Radicals U1V Roots
U5V Domain of a Radical We use the idea of roots to reverse powers. Because 32 = 9 and (-3)2 = 9, both 3 andExpression or Function
-3 are square roots of 9. Because 24 = 16 and (-2)4 = 16, both 2 and -2 are fourth
roots of 16. Because 23 = 8 and (-2)3 = -8, there is only one real cube root of 8 and
only one real cube root of -8. The cube root of 8 is 2 and the cube root of -8 is -2.
nth Roots
If a = bn for a positive integer n, then b is an nth root of a. If a = b2, then b is a
square root of a. If a = b3, then b is the cube root of a.
If n is a positive even integer and a is positive, then there are two real nth roots of
a. We call these roots even roots. The positive even root of a positive number is called
the prin ...
5.1 Introduction 5.2 Ratio And Proportionality 5.3 Similar Polygons 5.4 Basic Proportionality Theorem 5.5 Angle Bisector Theorem 5.6 Similar Triangles 5.7 Properties Of Similar Triangles
Similar to Roots and co efficeint of quadratic equation (20)
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This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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5. Quadratic Equation
£ It is the second degree equation, also said polynomial of
degree two.
£ The quadratic equation is derived from Latin language.
£ In Latin “Quadratus” means “Square”
6.
7. “The root of a number x is another number, its mean when a
number is multiplied to itself it gives number of times, equal x”.
Example:
The second root of 25 is 5, because 5 x5 = 25.
‡ The second root is called square root
‡ The third root is called cube root
8.
9. We know that and are the roots of the quadratic equation
where a, b are the co-efficient of and respectively, where
C is the constant term.
10. Relationship between roots and
quadratic equation
Let we consider and
then we find the sum and product of the roots.
We add both quadratic formulas alpha and beta
= +
Derivation
Relationship between roots and
quadratic equation
Sum of
roots
13. If we denote the sum and product of roots by
S and P respectively, then
14. œ The sum of the roots of the quadratic equation is
equal to the negation of the co-efficient of the
second term, dividing by the leading co-efficient.
œ The product of the roots of the quadratic equation is
equal to the constant term( the third term), divided by the
leading co-efficient.
16. Question No: 1
Find the relation between roots and co-efficient of a quadratic equation
without solving the equation
find sum and product
Solution:
Let be the roots of the given question.
Then,
17. Question No, 2
Find the relation when roots are connected by given relation.
Find relation between co-efficient of a, b and c of quadratic equation.
If alpha and beta are the roots of the equation
Find the values of the following;
(i)
(iv)
18. The given equation is
according to the problem alpha and Beta are the roots of the
equation
Therefore;
(i)
