Chapter 4: Quadratic Equations

Quadratic equations can be challenging for many students, but with practice and a clear understanding of the concepts, they can become easier to solve. In this document, we will provide you with practice questions and their solutions to help you improve your understanding of quadratic equations.

Introduction to Quadratic Equations
Quadratic equations are equations of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions to these equations are called the roots or zeros. Quadratic equations can have either two real roots, one real root, or complex roots.

Practice Questions:
1. Solve the quadratic equation: 2x^2 + 5x - 3 = 0.
2. Find the roots of the equation: x^2 - 9 = 0.
3. Solve the quadratic equation: 3x^2 + 7x + 2 = 0.
4. Find the roots of the equation: 4x^2 - 4x + 1 = 0.
5. Solve the quadratic equation: 6x^2 + 11x - 35 = 0.

Solutions:
1. To solve the equation 2x^2 + 5x - 3 = 0, we can factor it as (2x - 1)(x + 3) = 0. Therefore, the roots are x = 1/2 and x = -3.
2. The equation x^2 - 9 = 0 can be factored as (x - 3)(x + 3) = 0. Hence, the roots are x = 3 and x = -3.
3. By factoring the equation 3x^2 + 7x + 2 = 0, we get (3x + 1)(x + 2) = 0. Thus, the roots are x = -1/3 and x = -2.
4. The quadratic equation 4x^2 - 4x + 1 = 0 cannot be factored easily. Therefore, we can use the quadratic formula x = (-b ± √(b^2 - 4ac))/(2a) to find the roots. Plugging in the values, we get x = (4 ± √(16 - 16))/(8), which simplifies to x = 1/2. Hence, the equation has one real root.
5. By factoring the equation 6x^2 + 11x - 35 = 0, we obtain (2x - 5)(3x + 7) = 0. Thus, the roots are x = 5/2 and x = -7/3.

Conclusion
Practicing quadratic equations is essential to gain proficiency in solving them. By understanding the concepts and using various methods like factoring or the quadratic formula, you can solve these equations effectively. Remember to practice regularly and seek help when needed.