Quadratic functions and equations are described. A quadratic function is defined as f(x) = ax^2 + bx + c, where a ≠ 0. The graph of a quadratic function is a parabola. A quadratic equation is an equation of degree 2 in the form ax^2 + bx + c = 0, where a ≠ 0. There are three methods for solving quadratic equations: factorizing, completing the square, and the formula method. Factorizing involves factoring the left side of the equation into two binomial factors. Completing the square involves rewriting the equation into the form (x + p)^2 = q and then taking the square root of both sides.

1. Quadratic functions and equations
Quadratic functions: A function f given as
f(x) = a 𝑥2
+ b x + c, a ≠0, is called a
quadratic function. The graph of quadratic
function is a parabola.
e.g. 1) f(x) = 𝑥2
− 4x + 5.
2) f(x) = 3 𝑥2
− 9 x + 13
are quadratic functions.

2. Quadratic equations
An equation of degree 2 is called quadratic
equations.
General from: a x2 + b x + c = 0 where a, b,
c, ∈ R and a ≠ 0. The values of x satisfying
the quadratic equation are called roots of
the equation.

3. Quadratic equations
Examples: 1) 3x2 + x + 5 = 0,
2) -x2 + 7x + 5 = 0,
3) x2 + x = 0
are the quadratic equations.

4. Methods for solving the quadratic equations
There are three Methods of solving the
quadratic equations,
1) Factorizing of Quadratic Equation
2) Method of Completing the Square
3) Formula Method of Finding Roots

5. Factorizing of Quadratic Equation
Ex. Solve the following Quadratic Equations
1) x2 + 5x + 6 = 0
Sol.: x2 + 5x + 6 = 0
x2 + 2x + 3x + 6 = 0
x(x + 2) + 3(x + 2) = 0
(x + 2)(x + 3) = 0
Thus the two obtained factors of the quadratic
equation are (x + 2) and (x + 3)
.i.e. x =-2 & x = -3.

6. 2) x2 - 2x - 24 = 0
Sol.: x =-4 & x = 6.

7. Method of Completing the Square
Ex.1) Solve x2 + 4x + 4 =0
Sol: x2 + 4x + 4 =0
(x+2)2 = 0
(x+2)(x+2) =0
i.e. x= -2, -2

8. Ex.2) Solve x2 + 8x + 16 =0

9. THANK YOU