In today’s session I am going to tell you about the standard form in mathematics. Here we are going to discuss about the standard forms of quadratic equations and quadratic functions. Before I start telling you about Standard Form of quadratic equation, I think we must know about what the quadratic equations are. A quadratic equation is basically a polynomial equation that is uni-variate and of second degree. So a quadratic equation has a form ax^2 + bx + c = 0, and here a is not equal to 0. This form of the equation is known as the standard form of the quadratic equation. Here the most important thing about standard form is that it is not unique, means it can also be written in different way. For instance we have, 8x^2 + 8x + 16 = 0 is a standard form but it can also be written as x^2 + x + 2 = 0.
Now here we have some problems related to the standard form of the quadratic equation.
1. Write the given equation: 2x = 4x^2 + 2 into standard form.
We have equation 2x = 4x^2 + 2
4x^2 – 2x + 2 = 0
2x^2 – x + 1 = 0 this is the standard form of the given equation.
Similarly we have one more problem.
2. Given equation is 4x – 4 = 7x, change it in its standard form.
The answer is that the given equation is not a quadratic equation so we cannot change it in its standard form.
So here we learnt about the standard form of the quadratic equation, now we will learn about quadratic functions.
A quadratic function is a polynomial function which has a standard form
f(x) = a(x – m)^2 +n
The graph of the quadratic function is U in shape and this shape is known as parabola in mathematics. To get the x-intercept and y-intercept of the graph we just change the values of all the three coefficients which are m, n, a.
In the above equation the term (x – m)^2 is either be positive or zero, because it is a square term i.e. (x – m)^2 > = 0.
In the case if we multiply a on both the (left and right) sides of the equation then the value of a will be either positive or negative. So here we have two possibilities:
1. Coefficient a is negative: a(x – m)^2
So add n on both the sides
a(x -m)^2 + n
Here a(x -m)^2 + n represents the function f(x) and f(x)
2. If coefficient a is positive: a(x – m)^2 >= 0
a(x -m)^2 + n >= n
Here f(x) >= n so n is the minimum value of the quadratic function f(x).