Unit 2 Theory of Quadratic Equations Essential Guide

In Unit 2 Theory of Quadratic Equations, students will gain a comprehensive understanding of key algebraic principles related to quadratic equations. This unit covers essential concepts such as the discriminant, which helps determine the nature of the roots, and how to use these concepts to solve quadratic equations. Additionally, the unit explores the relationships between the coefficients and roots, including symmetric functions and the methods for forming quadratic equations. By the end of this unit, students will also master cube roots of unity, synthetic division, and applying these methods in practical scenarios, preparing them for advanced topics in algebra and problem-solving.

Exercise 2.1, Exercise 2.2 , Exercise 2.3, Exercise 2.4,
Exercise 2.5, Exercise 2.6, Exercise 2.7, Exercise 2.8

Multiple Choice Questions Test for Unit 2

To reinforce your learning of Unit 2 Theory of Quadratic Equations, we’ve included multiple-choice questions that cover a wide range of concepts. These questions test your knowledge of the discriminant, the nature of roots, the sum and product of roots, synthetic division, and the properties of cube roots of unity. Taking the quiz will not only evaluate your understanding but also provide you with an opportunity to apply these important concepts in a variety of problems. The MCQs are designed to help you build confidence and mastery over the topics covered in this unit.

Solutions of Unit 2 Theory of Quadratic Equations Key Points

1. The discriminant of the quadratic expression \[(ax^2 + bx + c)\] is given by \[(b^2 – 4ac)\]
2. Find discriminant of a given quadratic equation.
3. Discuss the nature of roots of a quadratic equation through discriminant.
4. Determine the nature of roots of a given quadratic equation and verify the result by solving the equation.
5. Determine the value of an unknown involved in a given quadratic equation when the nature of its roots is given.
6. Find cube roots of unity.
7. Recognize complex cube roots of unity as \[( \omega ) and ( \omega^2 )\]
8. Prove the properties of cube roots of unity.
9. Use properties of cube roots of unity to solve appropriate problems.
10. Find the relation between the roots and the coefficients of a quadratic equation.
11. Find the sum and product of roots of a given quadratic equation without solving it.
12. Find the value(s) of unknown(s) involved in a given quadratic equation when:
    • Sum of roots is equal to a multiple of the product of roots,
    • Sum of the squares of roots is equal to a given number,
    • Roots differ by a given number,
    • Roots satisfy a given relation (e.g., the relation \[2(\alpha) + 5 (\beta) = 7, where (\alpha) and (\beta)\] are the roots of the given equation.
    • Both sum and product of roots are equal to a given number.
13. Define symmetric functions of roots of a quadratic equation.
14. Evaluate a symmetric function of the roots of a quadratic equation in terms of its coefficients.
15. Establish the formula \[x^2 – (\text{sum of roots}) \, x + (\text{product of roots}) = 0\]
16. To find a quadratic equation from the given roots.

17. Form the quadratic equation whose roots, for example, are of the type:
\(2\alpha + 1, \, 2\beta + 1\)
\[
\alpha^2, \, \beta^2
\]
\[
\frac{1}{\alpha}, \, \frac{1}{\beta}
\]
\[
\alpha \beta, \, \frac{\beta}{\alpha}
\]
\[
\alpha + \beta, \, \frac{1}{\alpha} + \frac{1}{\beta}
\]

where \[(\alpha, \beta)\] are the roots of a given quadratic equation.

18. Describe the method of synthetic division.

19. Use synthetic division to

• find quotient and remainder when a given polynomial is divided by a linear polynomial,
• find the value(s) of unknown(s) if the zeros of a polynomial are given,
• find the value(s) of unknown(s) if the factors of a polynomial are given,
• solve a cubic equation if one root of the equation is given,
• solve a biquadratic (quartic) equation if two of the real roots of the equation are given.

20. Solve a system of two equations in two variables when

• one equation is linear and the other is quadratic,
• both the equations are quadratic.

21. Solve the real-life problems leading to quadratic equations.

Have a look at the Important Formulas of Unit 2 Theory of Quadratic Equations.

Solutions of Unit 3

By completing Unit 2 Theory of Quadratic Equations, you will have developed a strong understanding of quadratic equations, their properties, and their applications. Whether solving equations, analyzing the nature of roots, or applying the cube roots of unity, these skills will form the foundation for solving more complex algebraic problems.

With consistent practice and testing, you will gain the necessary tools. These tools will help you confidently approach and solve quadratic equations. You will be able to apply these equations in real-life situations. This will ensure that you have a solid understanding of algebraic principles. These principles will serve you well in future mathematical studies. For further assistance here is a complete playlist you YouTube lectures of unit 2 Theory of Quadratic Equations.