Standard Form of Quadratic Equation
2026-02-28 23:59 Diff
https://brightchamps.com/en-us/math/algebra/standard-form-of-quadratic-equation

To convert vertex form to standard form, we simplify\((x - h)^2 = (x - h)(x - h) \) Let’s see how to convert with an example, converting 2(x + 3)² - 5

Here, a = 2

h = -3

k = -5

\(2(x + 3)^2 - 5 = 0 \)

\(2(x + 3)(x + 3) - 5 = 0 \)

\(2(x^2 + 6x + 9) - 5 = 0 \)

\(2x^2 + 12x + 18 - 5 = 0 \)

\(2x^2 + 12x + 13 = 0 \)

How to Convert Standard Form of Quadratic Equation into Intercept Form? 


The quadratic equation in intercept form is a(x - p)(x - q) = 0, where (p, 0) and (q, 0) are the x-intercepts. To convert a standard form to an intercept form, we first find the roots of the quadratic equation, as p and q are the roots of the quadratic equation. Let’s learn it with an example,

For example, converting the quadratic equation x2 - 7x + 12 = 0 into intercept form
We first find the root of the quadratic equation.

\(x2 - 7x + 12 = 0 \)
Here, a = 1

b = -7

c = 12

To find the value of x we use quadratic equation:

\( x = \frac{7 \pm \sqrt{49 - 48}}{2} x = \frac{7 \pm \sqrt{1}}{2} x = \frac{7 \pm 1}{2} \text{So, } x = \frac{7 + 1}{2} \Rightarrow \frac{8}{2} = 4 x = \frac{7 - 1}{2} \Rightarrow \frac{6}{2} = 3 \)

As x = 4 and x = 3

Therefore, p = 4 and q = 3

The intercept form of the quadratic equation is:

\(a(x - p)(x - q) = 0 \)

Substituting the value of p and q:

\(1(x - 4)(x - 3) = 0 \)

How to Convert Intercept Form to Standard Form?
 

To convert a quadratic equation in intercept form to standard form, we simply use the intercept form. In other words, by simplifying (x - p)(x - q) = 0. 

For example, convert (2x + 3)(x -4) = 0 into standard form
\((2x + 3)(x - 4) = 2x^2 - 8x + 3x - 12 = 2x^2 - 5x - 12 \)

How to Represent Quadratic Functions in Standard Form in Graph?


The standard form of a quadratic function is f(x) = ax2 + bx + c, where a ≠ 0. The curve in the graph of a quadratic function is a parabola. 
 

  • Curve is a parabola
     
  • Width and slope depend on a
     
  • Vertex is at axis of symmetry
     
  • Parabola opens upwards if a>0, downwards if a<0