Quadratic Formula

A quadratic equation is a type of polynomial equation where the highest degree of the variable is 2. These equations play a critical role in mathematics, especially algebra, and are used in real-world scenarios like calculating areas, projectile motion, and solving business optimization problems.

The general form of a quadratic equation is:- 

ax² + bx + c = 0

Table Of Content

• Standard Form of a Quadratic Equation using Quadratic Formula
• How to Solve Quadratic Equations using Quadratic Formula
• Quadratic Formula
• Solved Examples of Quadratic Formula
• Applications of Quadratic Formual
• Fun Facts about Quadratic Formula
• Common Misconceptions of Quadratic Formula
• Practice Questions on Quadratic Formula

Standard Form of a Quadratic Equation using Quadratic Formula

The standard form of a quadratic equation is: ax² + bx + c = 0, where a ≠ 0

• If a = 0, it becomes a linear equation.

• The variable x must be raised to the power of 2, and the equation must include only one variable.

• This form is essential for applying formulas and solving questions.

How to Solve Quadratic Equations using Quadratic Formula

There are four primary methods used to solve a quadratic equation:

1. Factoring

Convert the equation into a product of binomials and solve each factor.

Example:
x² - 5x + 6 = 0
(x - 2)(x - 3) = 0
x = 2 or x = 3

 

2. Completing the Square

Transform the equation into a perfect square and solve.

Example:
x² - 4x + 4 = 0
(x - 2)² = 0
x = 2

 

3. Using the Quadratic Formula

This is the most versatile method, suitable for all types of quadratics:

x = [-b ± √(b² - 4ac)] / 2a

Here,

• The term under the square root (b² - 4ac) is called the discriminant (D).

• If D > 0 → Two real and distinct roots

• If D = 0 → Two real and equal roots

• If D < 0 → Complex roots
  
  

 

4. Taking Square Roots

For equations in the form x² = k

Example:
x² = 49
x = ±7

 

Quadratic Formula

To solve any quadratic equation:
ax² + bx + c = 0, use:

x = [-b ± √(b² - 4ac)] / 2a

This formula gives two solutions (roots), and works even when factoring is difficult.

 

Solved Examples of Quadratic Formula

Example 1:

Solve: x² - 6x = 16
Rewriting: x² - 6x - 16 = 0
By factoring: (x - 8)(x + 2) = 0
x = 8 or x = -2

 

Example 2:

Solve: x² - 16 = 0
Using identity: (x - 4)(x + 4) = 0
x = 4 or x = -4

 

Example 3:

Solve: y² = -2y + 2
Rewriting: y² + 2y - 2 = 0
Using quadratic formula:
y = [-2 ± √(4 + 8)] / 2 = [-2 ± √12]/2
y = -1 ± √3

 

Example 4 (Application-Based):

A rectangle has an area of 336 cm². Its length is 4 more than twice the width. Find the width.
Let width = x cm.
Then, length = 2x + 4
Area = length × width = x(2x + 4) = 336
2x² + 4x - 336 = 0
Divide by 2: x² + 2x - 168 = 0
Factoring: (x + 14)(x - 12) = 0
x = 12 (width)

 

Applications of Quadratic Formula

Quadratic equations are used to solve:

• Area and perimeter problems in geometry

• Projectile motion in physics

• Optimization problems in business and economics

• Speed, distance, and time word problems

Real-life example:

Calculating the maximum height a ball reaches when thrown upward follows a quadratic pattern.

 

Fun Facts about Quadratic Formula

• The word "quadratic" comes from the Latin word "quadratus", meaning square.

• Quadratic equations were studied as early as 2000 BC in Babylonian mathematics.

• The parabola, the graph of a quadratic, is also the shape of satellite dishes and suspension bridge cables.

 

Common Misconceptions of Quadratic Formula

• Misconception 1: Quadratic equations always have real solutions.
    Truth: Some have complex (imaginary) roots if the discriminant is negative.

• Misconception 2: Only 1 method exists to solve a quadratic.
    Truth: There are 6, and the method depends on the equation’s form.

• Misconception 3: The solution is always the point where the graph touches the x-axis.
    Truth: If the roots are imaginary, the graph never touches the x-axis.

 
Practice Questions on Quadratic Formula

1. Solve x² + 2x + 1 = 0

2. Solve 5x² + 6x + 1 = 0

3. Solve 2x² + 3x + 2 = 0

4. Solve x² - 4x + 6.25 = 0

5. Find the roots of x² - 49 = 0

Master the concepts and applications of the quadratic equation with expert-guided resources at Orchids International.