# Higher Algebra Hall & Knight The Theory of Quadratic Equations (Chapter 9) Solutions

Hall and Knight Higher Algebra Solutions for Chapter 9 ‘The Theory of Quadratic Equations’ comprises all the essential topics which help in forming a stronger command of the fundamentals of notation along with the explanation of roots, determination of variables, conditions for roots, magnitude and coefficient of roots. These solutions cover vital ideas regarding quadratic functions, real linear factors and radicals. You will also learn about the concepts of rational factors, real roots, integral functions, independent variables and linear function with the help of our solutions.

Higher Algebra By Hall and Knight ‘Theory of Quadratic Equations’ solutions include 3 exercises and 54 questions. These questions revolve around all the fundamentals that are included in the chapter and provide a very convenient way of practising the chapter thoroughly. Your engagement with these exercises and the chapter at large will enable you to tackle all the possible kinds of questions that can be formed from the chapter for competitive exams like IIT JEE and NEET. You can then become familiar with the ways of deriving solutions and managing time.

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## Important Topics for Hall and Knight Higher Algebra Chapter 9: The Theory of Quadratic Equations

**Quadratic Equations **

You would have learnt about quadratic equations in your earlier classes, in this chapter you will get to revise your basics and take them to an advanced level. You will encounter questions where you will have to prove some of the important propositions connected with the roots and coefficients of the equations. You will also learn about the limitations of a quadratic equation, one of which is that no quadratic equation can have more than two roots.

**Condition-based problems **

In this chapter, you will come across different kinds of questions where you will have to work according to given conditions. There are terms like linear factors, rational factors, real roots and range-based values which have been included in the chapter to guide the discourse of the questions.

### Exercise Discussion of Hall and Knight Higher Algebra Chapter 9: The Theory of Quadratic Equations

**Example IX a **

**This exercise contains a total of 30 questions.**

**The exercise begins with simple questions where you are provided with two values and you are required to form equations in such a way that their roots turn out to be the given values. The first 12 questions of the exercise belong to this category.**

- Then there are problems where you are provided with two equations and you are required to prove that their roots are real.

- There are some questions where you are provided with equations that have a variable in them and with the help of the given equation and the roots of that equation provided in the question, you are required to find the value of the variable.

- There are similar questions where you need to find the value of the variable in the quadratic equation under the condition that the roots of the equation are real.

- There are some problems where you are required to prove that the equations are rational

- There are questions with the condition that the roots are equal in magnitude but opposite in sign.

- The last portion of the exercise has questions that discuss the roots of the equation as squares and cubes, deal with conditions and signs of the roots.

**Example IX b **

**The exercise has a total of 14 questions.**

**The exercise begins with questions where you are required to determine the limits between the values must lie such that the equation has real roots.**

- There are questions where you are provided with roots of the given equations and you are required to find the values of variables included in the equation

- There are questions where you have to determine the roots and solve the equation according to the given ratio

- You are also supposed to find the greatest and the lowest value of the variables in case of real values of the roots.

**Example IX c **

**There are a to**tal of 10 questions in the exercise.- It begins with a question which requires you to find the value of m given in the expression, such that it is capable of resolution into two rational factors.
- There are questions where the equations are equivalent to the product of two linear factors and accordingly you are required to find the variable.
- There are questions where you have two variables in the given equation which are real quantities and you are also provided with the range of their values, according to which you are required to solve the problem.
- The exercise majorly comprises questions which deal with different conditions regarding real quantities, rational factors and linear factors.

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