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
• a is the coefficient of x²
• b is the coefficient of x
• 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: Determined by the highest exponent of x. 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. 
 

Derivation of the Quadratic Equation (Standard Form)

A quadratic equation represents a parabola and is generally written in the form:
 

[ax² + bx + c = 0]
 

Let’s derive this form from the factored (intercept) form of a quadratic equation.

Step 1: Start with the Intercept Form

Let the quadratic equation have roots ( p ) and ( q ).
Then, the equation can be written as:

[y = a(x - p)(x - q)]

where ( a ) is a non-zero constant (the leading coefficient).

Step 2: Expand the Expression

Multiply the two brackets:
 

[y = a[(x - p)(x - q)]]

[y = a[x² - (p + q)x + pq]]

Step 3: Simplify

Distribute ( a ) to each term:

[y = ax² - a(p + q)x + apq]

Step 4: Compare with the Standard Form

The general standard form of a quadratic equation is:

[y = ax² + bx + c]

Comparing both, we get:

[b = -a(p + q), c = apq]

Step 5: Convert to Equation Form

To write it as a quadratic equation (not function), set ( y = 0 ):

[ax²+ bx + c = 0]

Hence, the standard form of a quadratic equation is:

[{ax² + bx + c = 0}]

where

• ( a not equal to 0 )
   
• ( b = -a(p + q))
   
• ( c = apq )

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

Step 1: 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


Step 2: Comparing the coefficients of x on both sides:

bx = -2·a·x·h

\( h = \frac{-bx}{2ax} \)

h = \( -\frac{b}{2a} \), let’s consider this equation as (1)
 

Step 3:Comparing the constants on both sides
c = ah² + k
 

Step 4: 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) = -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:

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

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) = 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. 
 

• 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

Mastering the standard form of a quadratic equation becomes easy with the right strategies.
These practical tips will help you solve equations faster, avoid common mistakes, and understand real-life applications effectively.

 

• Understand the formula deeply: Always remember the standard form ax² + bx + c = 0. Knowing what each coefficient represents helps in quick identification of a, b, and c.
  
   
• Check for simplification: Before solving, simplify the equation by dividing through by common factors so a, b, and c are smaller and easier to work with.
  
   
• Learn the discriminant rule: Use D = b² − 4ac to quickly identify the nature of roots — positive (two real roots), zero (one real root), or negative (complex roots).
  
   
• Watch the signs carefully: Many mistakes happen due to sign errors when substituting values. Double-check negatives and brackets.
  
   
• Factor when possible: Before using the quadratic formula, check if the equation can be easily factored. It saves time and minimizes calculation errors.

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.
  
   
• Quadratic equations are used to design lenses and mirrors in optics. For example, parabolic mirrors in telescopes and headlights use quadratic curves to focus light precisely.
  
   
• Quadratic equations help in modeling cost and profit functions to determine the break-even points where revenue equals expenses, aiding in strategic business decisions.