Standard Form of Quadratic Equation

The standard form of a quadratic equation is ax2 + bx + c = 0, where 
x is the variable, a ≠ 0, 
a, b, and c are real numbers. 
Here, a is the coefficient of x²
b is the coefficient of x
And c is the constant

There are three forms to write the quadratic equation, including:

• Standard form: ax² + bx + c = 0
• Vertex form: a(x - h)² + k = 0
• Intercept form: a(x - p)(x - q) = 0
   

A second-degree polynomial equation is the quadratic equation, and its standard form is ax² + bx + c = 0. The characteristics of the standard form of quadratic equations are:

•  Degree of equation: The degree of the equation is determined using the highest value of the exponent. In a quadratic equation, the highest power of x is 2. So the quadratic equation is a second-degree polynomial. 

• Shape of the graph: The graph of a quadratic equation forms a parabola; the shape of the parabola depends on the value of a. If a > 0, then the parabola opens upwards, and if a < 0, then the parabola opens downwards.

• Vertex of parabola: The vertex of the parabola represents the minimum or maximum point of the graph, depending on the sign of a. It is located at (-b/2a, f (-b/2a)), where f(x) is the quadratic expression. 

• Axis of symmetry: The axis of symmetry is the vertical line that passes through the vertex, dividing the parabola into halves. The vertical line is given by x = -b/2a.
   

The standard form of a quadratic function is written as:
f(x) = ax² + bx + c = 0
Here, a, b, and c are the constant coefficients, and x is the variable. It is also known as the second-degree equation. In a quadratic function, the value of a ≠ 0, because if the value of a is 0, then the function will not be quadratic, as the highest degree in a quadratic is 2. 
 

The standard form of a quadratic equation is ax² + bx + c = 0, and the vertex form is a(x - h)² + k = 0, where (h, k) are the vertices of the quadratic function. To convert the equation from standard form to vertex form, we compare these two equations:
ax² + bx + c = a(x - h)² + k
Substituting the value of (x - h)² in the equation, (x - h)² = x² - 2xh + h²
ax² + bx + c = a (x² - 2xh + h^j) + k
ax² + bx + c = ax² -2axh + ah² + k
Comparing the coefficients of x on both sides:
bx = -2axh
h = -bx/2ax
h = -b/2a, let’s consider this equation as (1)

Comparing the constants on both sides
c = ah² + k

Substituting the value of h from (1)
c = a(-b/2a)² + k
c = a(b²/4a²) + k
c = (b²/4a) + k
c - (b²/4a) = k
k = c - (b2/4a)

Therefore, we can use the formulas h = -b/2a and k = c - (b²/4a) to convert a standard form of a quadratic equation into vertex form. 

For example, convert 3x² + 6x - 5 = 0 to vertex form
Here, a = 3
b = 6
c = -5

Given, equation is a(x - h)2 + k = 0
Finding the value of h and k:
h = -b/2a
h = -(6/2 × 3)
= -6/6 
= -1

k = c - (b²/4a)
= -5 - (62/4 × 3)
= -5 - (36/12)
= -5 - 3 
= -8

Substituting the value of h and k in: a(x - h)² + k = 0
 3(x - (-1))² - 8 = 0
3(x + 1)² -8 
 

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 (x2 + 6x + 9) -5 = 0
2x2 + 12x + 18 - 5 =0
2x2 + 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 = (-b ± (√b2 - 4ac)2a
x = (-(-7) ± (√(-7)2 - (4 × 1 × 12)/2
x = (7 ± (√49 - 48))/2
x = (7 ± √1)/ 2
x = (7 ± 1)/2
So, x = (7 + 1)/2 ⇒ 8/2 = 4
x = (7 -1)/2 ⇒ 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) = 2x2 - 8x + 3x - 12 = 2x2 - 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. 
The width and slope of the parabola depend on the value of a. The vertex of the parabola is where the axis of symmetry crosses, and as it has the axis of symmetry, all parabolas are symmetric. The parabola opens upwards, and it opens downwards if a is negative. 
 

In real-life, we use the standard form of a quadratic equation, where the relationships involve squared terms. The few applications of the standard form of quadratic equations

• To calculate the maximum height, range, and landing point of the object in projectile motion such as basketball shot and thrown rescue rope we use the quadratic equations. For example, in sports quadratic equations analyze the trajectory of a football to see the player's performance. 

• Companies use the quadratic equation in profit modeling. For example, to predict the optimal number of products to produce and the best price to maximize revenue, quadratic equations are required.

• In building bridges and arches, quadratic equations are used to distribute the weight and prevent collapse. For example, to ensure stability and prevent structural failure.