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

• What is a Quadratic Equation?
• Standard Form of a Quadratic Equation
• How to Solve Quadratic Equations
• Quadratic Formula
• Solved Examples of Quadratic Equations
• Applications of Quadratic Equations
• Fun Facts about Quadratics
• Common Misconceptions
• Practice Questions
• Conclusion
• FAQs on Quadratic Equations

 What is a Quadratic Equation?

A quadratic equation is defined as a second-degree polynomial equation in a single variable. The highest power of the variable x is 2, which makes it a quadratic.

Example:

• x² + 5x + 6 = 0

• 3x² – 4 = 0

• (x – 3)(x + 2) = 0
  
  

Quadratic equations always have two solutions (or roots), which may be real or complex depending on the discriminant value.

 

Standard Form of a Quadratic Equation

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 methods consistently.
  
  

How to Solve Quadratic Equations

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

Best 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 Equations

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 Equations

Quadratic equations are used to solve:

• Area and perimeter problems in geometry

• Projectile motion in physics

• Optimization 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 Quadratics

• 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

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

• Misconception 2: Only one method exists to solve a quadratic.
    Truth: There are four - 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

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
   
   

Conclusion

A quadratic equation is more than just a formula - it’s a powerful tool used in mathematics and real-world situations alike. From factoring to applying the quadratic formula, mastering each method of solving quadratic equations is essential for progressing in algebra and higher mathematics.

Understanding the roots of quadratic equations, interpreting discriminants, and applying formulas can open doors to solving geometric, physical, and statistical problems effectively. With a strong grasp on quadratics, you're well-prepared for both academic and competitive success.

 

 Frequently Asked Qs on Quadratic Equations

Q1. What is a quadratic equation?

Answer. It is a second-degree polynomial equation in the form ax² + bx + c = 0 where a ≠ 0.

Q2. What are the methods to solve a quadratic equation?

Answer.Factoring, Completing the Square, Using Square Roots, and the Quadratic Formula.

Q3. Can a quadratic equation have only one root?

Answer. Yes, when the discriminant is 0, both roots are equal.

Q4. What is the discriminant of a quadratic equation?

Answer. It is D = b² – 4ac. It tells us the nature of the roots.

Q5. What are real-life uses of quadratic equations?

Answer. Solving motion problems, maximizing profit, calculating areas, and more.

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