1. Introduction to Quadratic Equations: - A quadratic equation is a second-degree polynomial equation in a single variable x, with the highest power of x being 2. The general form of a quadratic equation is: -
ax^2 + bx + c = 0
where: - a, b, and c are constants,
a not equal to zero and x is the variable to be solved.
Quadratic equations arise frequently in various mathematical and real-world problems, such as physics, engineering, economics, and more.
2. Forms of Quadratic Equations: -
(i) Standard form: - ax^2 + bx + c = 0, where a, b, and c are constants.
(ii) Factorized form: - a(x - p)(x - q) = 0, where p and q are the roots of the equation.
(iii) Vertex form: - a(x - h)^2 + k = 0, where (h, k) is the vertex of the parabola.
3. Solutions of Quadratic Equations: -
(i) Factorization method: - If a quadratic equation can be factored, the roots can be found by setting each factor equal to zero.
Example: - x^2 - 5x + 6 = 0
(x - 2)(x - 3) = 0
Roots are x = 2 and x = 3.
(ii) Quadratic formula: - If the quadratic equation cannot be factored easily, the roots can be found using the quadratic formula (shri Dharacharya’s Formula): - x = -b plus minus sqrt{b^2 - 4ac}/{2a} Here, b^2 - 4ac is called the discriminant (D).
Based on the discriminant, we can determine the nature of the roots:
- If D greater than 0, the equation has real, distinct and irrational roots.
- If D = 0, the equation has real and equal roots.
- If D less than 0, the equation has complex and distinct roots.
- If D greater than zero and a perfect square, the equation has real, distinct and rational roots.
(iii) Completing the square: - This method involves rewriting the quadratic equation in the form (x - h)^2 = k, and then solving for x. Example: - x^2 + 6x + 9 = 0
(x + 3)^2 = 0
X = -3, -3
4. Sum and product of roots: - For the quadratic equation ax^2 + bx + c = 0,
- Sum of the roots: - p+q = -b/a = -(Coefficient of x)/Coefficient of x^2.
- Product of the roots: - pq = c/a = Constant/Coefficient of x^2.
- If roots of quadratic equation are given then the quadratic equation is: - x^2- (sum of roots) + (product of roots) = 0.
5. Graph of a Quadratic Equation: - The graph of a quadratic equation y = ax^2 + bx + c is a parabola.
- If a is greater than zero, the parabola opens upwards.
- If a is less than zero, the parabola opens downwards.
- The vertex of the parabola is the point (-b/2a,-D/4a).
This basic introduction should provide students with a solid foundation to understand and solve quadratic equations.