Quadratics

Introduction to Quadratics

Topic of quadratics is important in mathematics and is used in real life too. The word 'quadratic' comes from the Latin word 'quadratus', which means 'square', because the highest power of the variable is 2 in a quadratic expression.

A quadratic expression is usually written as ax2+bx+c, where a, b, and c are constants and a≠0. The quadratic equation can be solved by methods such as factorisation, completing the square, or quadratic formulas. The graph of a quadratic equation is called a parabola, which can open upwards or downwards depending on the value of a.

In this article, we will learn about quadratics, their different methods to solve them, and real-world applications. Step-by-step solved examples are provided to help you build a clear understanding of this topic.

Table of Contents

• What is Quadratic Equation?
• Standard Form of Quadratic Equation
• Quadratics Formula
• Examples of Quadratics
• How to Solve Quadratic Equations?
• Factoring of Quadratics
• Completing the Square Method
• Using Quadratic Formula
• Taking the Square Root
• Solved Problems
• Applications of Quadratics
• FAQs on Quadratics

What is Quadratic Equation?

A quadratic equation is a type of polynomial equation in mathematics where the highest power of the variable is 2. It is usually written in the standard form. Quadratic equations are important because they help us solve problems related to areas, heights, motion, and many real-life situations.

Standard Form of Quadratic Equation

A standard form of a quadratic equation is written in the form: 

ax2+bx+c=0

Here:

• x is a variable

• a, b, and c are called coefficients (a≠0)

• The ax2 is called the quadratic term, bx is the linear term, and c is the constant term.

Quadratics Formula

The quadratic formula is used to find the roots (solutions) of a quadratic equation. Since a quadratic equation has degree 2, it will always have two solutions. It is usually written as:

ax2+bx+c=0 , where a≠0

To solve a quadratic, we use the quadratic formula:

x=−b±b2−4ac2a

Here:

• a = number before x2

• b = number before x

• c = constant number

The part inside the square root (b2−4ac) is called the discriminant (D).It tells us the nature of the solutions of the quadratic equation.

• If D > 0, we get 2 different real answers.

• If D = 0, we get one real answer.

• If D < 0, we get no real answer.

Examples of Quadratics

Below are examples of quadratic equations of the form ax² + bx + c = 0:

• x² + 3x – 10 = 0

• 2x² – 7x + 5 = 0

• 4x² + 6x + 9 = 0

• –3x² + 8x – 2 = 0

Examples of quadratic equations without a constant term (c = 0):

• x² – 4x = 0

• –2x² + 7x = 0

• 5x² – 15x = 0

• –6x² – 2x = 0

• 9x² + 12x = 0

• –4x² + x = 0

Examples of quadratic equations in factored form:

• (x – 7)(x + 2) = 0 → x² – 5x – 14 = 0

• –2(x – 3)(x + 4) = 0 → –2x² – 2x + 24 = 0

• (x – 8)(x – 1) = 0 → x² – 9x + 8 = 0

• (x + 5)(x – 2) = 0 → x² + 3x – 10 = 0

• (2x – 1)(x + 3) = 0 → 2x² + 5x – 3 = 0

• (3x + 2)(x – 4) = 0 → 3x² – 10x – 8 = 0

Examples of quadratic equations without the linear term (bx = 0):

• x² – 25 = 0

• 3x² + 12 = 0

• –5x² – 20 = 0

• 7x² – 49 = 0

• –2x² + 18 = 0

• 4x² – 81 = 0
  

How to Solve Quadratic Equations?

Quadratic equations are equations in which the highest power of the variable is 2. They are written as:

ax2+bx+c=0, where a≠0.

There are different ways to solve quadratic equations. Let us try them one by one in simple words.

1. Factoring of Quadratics

In this method, we divide the mid-term and write the quadratic Equation as a product of two factors.

Example: x2−5x+6=0

Here, we can write it as (x+2)(x+3)=0

So, x=−2 or x=−3

2. Completing the Square Method

In this method, we make a quadratic expression a perfect square.

• Example: x2+6x+5=0

• Take half of 6, square it, and add +9 on both sides:

• x2+6x+9=4

• Now, (x+3)2=4

• So, x+3=±2

• Therefore, x=−3+2=−1 or x=−3−2=−5.

3. Using Quadratic Formula 

If factoring or completing the square is difficult, we can use the formula directly:

x=−b±b2−4ac2a

Example: 2x2+3x−2=0

Here, A=2,B=3,C=−2

x=−3±32−4(2)−22(2)

x=−3±9+164

x=−3±54

So, x=24=12 or x=−84=−2

4. Taking the Square Root

If the quadratic equation is already square, we can easily solve it by taking square roots.

Example: x2=16

Take the square root of both sides: x=±4

Solved Problems

Problem 1: Solve the quadratic equation x2−5x+6=0

Solution: We need to factorize the equation.

x2−5x+6=(x−2)(x−3)=0

So, x−2=0 or x−3=0

That means x=2 or x=3.

Problem 2: Solve the quadratic equation x2+4x+4=0

Solution : 

x2+4x+4=(x+2)(x+2)=(x+2)2

So, (x+2)2=0

That means x=−2

Problem 3: Solve using the quadratic formula 2x2−3x−2=0

Solution:

The quadratic formula is x=(−b±(b2)−4ac)2a

Here, a=2,b=−3,c=−2

x=[−(−3)±((−3)2−4(2)(−2))]2(2)

x=3±9+164

x=3±254

x=3±54

Case 1: 3+54=84=2

Case 2: 3−54=−24=−12

Applications of Quadratics

Quadratic equations are not only in books, but they are also used in many real-life situations. Some examples are:

• Projectile motion: When you throw a ball in the air, its path is curved like a parabola. Quadratic equations are used to find how high it goes and where it will land.

• Area problems: Sometimes to find the maximum area of a rectangle or a box, quadratic equations are used.

• Profit and loss in business: Businesses use quadratics to find out how to make maximum profit or minimum loss.

• Physics and engineering: Many formulas in physics, like speed, height, and energy, use quadratic equations.

• Daily life: For example, when building bridges, designing roller coasters, or even calculating how long it takes for something to fall, quadratics are used.
  
  

FAQs on Quadratics

1. How do you explain quadratics?

Ans: In math, we define a quadratic equation as an equation of degree 2, meaning that the highest exponent of this function is 2. The standard form of a quadratic is y = ax²+ bx+ c , where a, b, and c are numbers and a cannot be 0. Examples of quadratic equations include all of these: y =x²+ 3x+ 1.

2. What math is quadratics in?

Ans: Quadratic Equation. Quadratic equations are second-degree algebraic expressions and are of the form ax²+ bx+ c = 0. The term "quadratic" comes from the Latin word "quadratus," meaning square, which refers to the fact that the variable x is squared in the equation.

3. Why is quadratic 2 and not 4?

Ans: The prefix “quad” means “four”, and quadratic expressions involve powers of x up to the second power, not the fourth power.

4. Are quadratics algebra 1 or 2?

Ans: Algebra 1 typically includes evaluating expressions, writing equations, graphing functions, solving quadratics, and understanding inequalities.

5. How to derive the quadratic formula?

Ans: The quadratic formula, x=−b±b2−4ac2a , can be derived by completing the square on the general standard form of a quadratic equation. Recall that completing the square is a method for solving quadratic equations.