Definition:
 Polynomials of degree two are called quadratic experssions/equations.
 Quadratic equations are of the form ax^{2} + bx + c = 0. Where a,b,c are constants and a <> 0, is in standard form for a quadratic equation.
 The letters a, b, and c are called coefficients: the quadratic coefficient a is the coefficient of x^{2}, the linear coefficient b is the coefficient of x, and c is the constant coefficient, also called the free term or constant term.
 Quadratic equations of type ax^{2} + bx + c = 0 and ax^{2} + bx = 0 (c is 0) can be factored to solve for x.
 If a = 0 then it become a linear equation.
 If the coefficient of second degree term is one (a=1) then the quadratic equation it is called a monic quadratic polynomial.
Example: x^{2}+4x=7, x^{2}6x+3,............etc, are called
monic quadratic expression.
Solution of Monic Quadratic Equations:
Example:
Factorize, x^{2}+7x+12.
Solution:
Given that, x^{2}+7x+12.
A quadratic monic polynomial in x is a perfect square if
the constant term in it is equal to the square of half the coefficient of x.
Therefore, if (12/2)^{2} = (6)^{2} = 36 is added to x^{2}+12x
it becomes a perfect square.
Therefore, we add and subtract 36 to given equation.
= x^{2}+7x+12 = (x^{2}+7x+36)36+27
(x^{2}+7x+36) this is form of (a^{2}+2ab+b^{2}) =
(a+b)^{2}.
Here a = x and b = 6, substitute these values in the formula, we get
= (x+6)^{2}36+27
= (x+6)^{2}9
=(x+6)^{2}3^{2}
This is in the form of a^{2}b^{2} = (a+b)(ab)
Here a = x+6 and b = 3, substitute these values in the formula, we get
= (x+6+3)(x+63)
= (x+9)(x+3)
Directions: Solve the following problems. Also write at least ten examples of your own.
