Quadratic functions are polynomial functions in algebra, and they contain one or more than one variables. The highest degree term in a quadratic function is of the second degree. Other names for quadratic equations are quadratic polynomial, polynomial of degree 2 and sometimes simply quadratic. For an example, a quadratic function can be: f (x, y, z) = ax2 + by2 +cz2 + dxy + exz + fyz + gx + hy + iz + j
In the above example, the variables are x, y and z, the function also contains some special terms like x2, y2, z2, xy, xz, yz and also contains one constant term which is j.
One example of univariate or one variable quadratic function is: f (x, y) = ax2 + bx + c , where a "≠" 0 and c is a constant.
A quadratic function that has only three terms is known as quadratic trinomial. The general form of a trinomial quadratic function is:
a x 2 + b x + c
Where, a, b and c are non-zero constants and x is the variable. In the quadratic function, a is known as the leading coefficient, b is known as linear coefficient and c is known as the additive constant.
The D or the discriminant of the quadratic trinomial is defined as:
D = b 2 − 4 a c
Examples of the quadratic trinomials are:
x 2 + 7 x + 6, 2 x 2 + 3 x – 4.
The first equation in the above given example is a quadratic trinomial what a leading coefficient 1.
Example 1: Solve the equation x2 + 5x + 6
In order to factor the above given equation, the first step is to find two numbers such that the multiplication of the two numbers is 6, which is the constant term in the equation), and the addition of the two numbers is 5, which is the coefficient of x.
So, the two numbers will be, 2 and 3. Since, 2 × 3 = 6 and 2 + 3 = 5.
After that the resultant numbers are added to x to form the two binomial factors which will be, (x + 2) and (x + 3).
The factorization of the trinomial function will be:
x2 + 5x + 6 = (x + 2)(x + 3)
Example 2: Solve the equation x2 − 5x + 6
In order to factor this equation, finding the numbers that will add to -5 and multiply to give the product of 6 is the first step.
The two numbers will be, -2 and -3. Since, (-2) × (-3) = 6 and (-2) + (-3) = -5.
The next step is to form the two binomial factors which will be, (x + (-2)) and (x + (-3)).
The factorization will be:
x2 − 5x + 6 = (x+ (-2)) (x + (-3))
= (x - 2) (x - 3)
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