In the standard form of the quadratic expression, only the terms x2, x along with constant numbers are expressed (Eq. 1). This makes the standard form of a quadratic expression a specific type of polynomial expression, whereby the shape is a parabola, as the highest exponent of any x term is 2, which gives the expression a parabolic shape, no matter what the other variables in the equation are. The x2 and x terms may also have coefficients attached to them, which are denoted as a, b and c.
Eq. 1: \(y = ax^2 + bx + c\)
Unlike the vertex and factored forms of the quadratic expression, the standard form of the expression does not directly correspond to the vertex of the parabola, or to its x-intercepts. This can be seen by changing the a (Figure 1), b (Figure 2) and c (Figure 3) values in the equation. Although these values affect the shape of the parabola, they do not have the obvious connection to either the position of the vertex, the size of the parabola, or its x-intercepts in the same way that the other forms of the quadratic expression do.
Figure 1: Changing the a value of the standard form of the quadratic expression. A change in the a value causes the shape of the parabola to become more stretched as it goes further away from 0, with positive values of a giving parabolas that open upwards, and negative values of a giving parabolas that open downwards.
Figure 2: Changing the b value of the standard form of the quadratic expression. As the b value diverges from 0, the parabola becomes more linear, but also steeper. This apparent linearity is only within the scale of the x values plotted, on a macro-scale, the parabola remains due to the x2 term.
Figure 3: Changing the c value of the standard form of the quadratic expression. As the value of c diverges from 0, the location of the parabola’s vertex changes. However, it does not directly correlate to the vertex's location as the k value in vertex form does.
Unlike the other forms of quadratic expression, the standard form generally (see below for an exception) does not contain any brackets. The standard form is expanded (which removes the brackets), and it has like terms combined, which collapses all the terms into only those terms found in the standard form, namely ax2, bx and c terms. All quadratic expressions can be presented in the standard form, and it can be readily converted to other forms of quadratic expressions. Furthermore, the quadratic formula (Eq. 2) can use used to rapidly solve expressions in the standard form.
Eq. 2: \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)
Sometimes the terms in the standard form are divisible by a greatest common factor. This means you can express the quadratic equation inside brackets, with a number on the outside of the brackets that all the terms in the standard form within the brackets are multiplied by (Eq. 3). This is still the standard form, but the brackets make it clearer for reading, and can also sometimes allow the form to be converted into the factored form easier.
Eq. 3: \(y = 2x^2 + 10x + 12 = 2(x^2 + 5x + 6)\)