Introduction to Quadratic Equations: Concepts, Methods, and Real-Life Applications

Q: Where are quadratic equations used?

<div class="fusion-layout-column fusion_builder_column fusion-builder-column-39 fusion_builder_column_1_1 1_1 fusion-flex-column" style="--awb-bg-size:cover;--awb-width-large:100%;--awb-margin-top-large:0px;--awb-spacing-right-large:1.92%;--awb-margin-bottom-large:20px;--awb-spacing-left-large:1.92%;--awb-width-medium:100%;--awb-order-medium:0;--awb-spacing-right-medium:1.92%;--awb-spacing-left-medium:1.92%;--awb-width-small:100%;--awb-order-small:0;--awb-spacing-right-small:1.92%;--awb-spacing-left-small:1.92%;"> Quadratic equations are used in physics, geometry, economics, engineering, and optimization problems.

Q: What are the methods to solve quadratic equations?

<div class="fusion-layout-column fusion_builder_column fusion-builder-column-42 fusion_builder_column_1_1 1_1 fusion-flex-column" style="--awb-bg-size:cover;--awb-width-large:100%;--awb-margin-top-large:0px;--awb-spacing-right-large:1.92%;--awb-margin-bottom-large:20px;--awb-spacing-left-large:1.92%;--awb-width-medium:100%;--awb-order-medium:0;--awb-spacing-right-medium:1.92%;--awb-spacing-left-medium:1.92%;--awb-width-small:100%;--awb-order-small:0;--awb-spacing-right-small:1.92%;--awb-spacing-left-small:1.92%;"> Factorization, completing the square, quadratic formula, and graphical method.

Introduction to Quadratic Equations

Quadratic equations are a crucial part of algebra, forming the foundation for solving complex problems across various disciplines. Let’s dive into what quadratic equations are, explore their methods of solution, and review real-life applications.

What is a Quadratic Equation?

A quadratic equation is any equation that can be expressed as:

$\boldsymbol{ax^2 + bx + c = 0}$

Where:

$\boldsymbol{a, b, c}$ are constants ( $\boldsymbol{a \neq 0}$ ).
$\boldsymbol{x}$ is the variable.

Standard Form of Quadratic Equations

The standard form of a quadratic equation is:

$\boldsymbol{ax^2 + bx + c = 0}$

Key components:

$\boldsymbol{ax^2}$ : The quadratic term.
$\boldsymbol{bx}$ : The linear term.
$\boldsymbol{c}$ : The constant term.

Example:

Convert $\boldsymbol{3x - 2x^2 + 5 = 0}$ into standard form:
Rearrange terms: $\boldsymbol{-2x^2 + 3x + 5 = 0}$ .
Multiply by $\boldsymbol{-1}$ : $\boldsymbol{2x^2 - 3x - 5 = 0}$ .

Methods to Solve Quadratic Equations

1. Factorization

This involves expressing the quadratic equation as a product of two binomials.

Example: Solve $\boldsymbol{x^2 - 7x + 10 = 0}$ .
Steps:

Factorize: $\boldsymbol{(x - 5)(x - 2) = 0}$ .
Solve: $\boldsymbol{x = 5}$ , $\boldsymbol{x = 2}$ .

Answer: $\boldsymbol{x = 5, x = 2}$ .

2. Completing the Square

This technique transforms the quadratic equation into a perfect square trinomial.

Example: Solve $\boldsymbol{x^2 + 6x + 5 = 0}$ .
Steps:

Rewrite: $\boldsymbol{x^2 + 6x = -5}$ .
Add $\boldsymbol{(6/2)^2 = 9}$ : $\boldsymbol{x^2 + 6x + 9 = 4}$ .
Factorize: $\boldsymbol{(x + 3)^2 = 4}$ .
Solve: $\boldsymbol{x + 3 = \pm 2}$ .
- $\boldsymbol{x = -1}$ , $\boldsymbol{x = -5}$ .

Answer: $\boldsymbol{x = -1, x = -5}$ .

3. Quadratic Formula

The quadratic formula is:

$\displaystyle\boldsymbol{x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}$

Example: Solve $\boldsymbol{2x^2 + 3x - 2 = 0}$ .
Steps:

Identify $\boldsymbol{a = 2, b = 3, c = -2}$ .
Discriminant: $\boldsymbol{b^2 - 4ac = 3^2 - 4(2)(-2) = 9 + 16 = 25}$ .
Apply formula:
$\displaystyle\boldsymbol{x = \frac{-3 \pm \sqrt{25}}{2(2)}}$
$\displaystyle\boldsymbol{x = \frac{6 \pm 12}{4}}$ .

Solutions: $\displaystyle\boldsymbol{x = \frac{1}{2}, x = -2}$ .

4. Graphical Method

In this method, the equation $\boldsymbol{y = ax^2 + bx + c}$ is plotted as a parabola. The roots are the points where the parabola intersects the x-axis.

Example: For $\boldsymbol{y = x^2 - 4x + 3}$ :

Plot $\boldsymbol{y = (x - 1)(x - 3)}$ .
Intersection points: $\boldsymbol{x = 1, x = 3}$ .

Answer: $\boldsymbol{x = 1, x = 3}$ .

Nature of Roots

The discriminant ( $\boldsymbol{\Delta = b^2 - 4ac}$ ) determines the type of roots:

$\boldsymbol{\Delta > 0}$ : Two distinct real roots.
$\boldsymbol{\Delta = 0}$ : One repeated root.
$\boldsymbol{\Delta < 0}$ : Complex roots.

Example: Solve $\boldsymbol{x^2 + 4x + 8 = 0}$ .

Discriminant: $\boldsymbol{\Delta = b^2 - 4ac = 16 - 32 = -16}$ .
Complex roots:
$\displaystyle\boldsymbol{x = \frac{-4 \pm \sqrt{-16}}{2}}$
$\boldsymbol{x = -2 + 2i, x = -2 - 2i}$ .

Answer: $\boldsymbol{x = -2 + 2i, x = -2 - 2i}$ .

Applications of Quadratic Equations

Projectile Motion: Calculating the path of an object in free fall.
Geometry: Finding dimensions or areas of geometric shapes.
Optimization: Maximizing profits or minimizing costs in economics.

Example:
A rectangle’s area is 24 m². If the length is 2 m more than the width, find the dimensions.

Let width = $\boldsymbol{x}$ . Length = $\boldsymbol{x + 2}$ .
Area: $\boldsymbol{x(x + 2) = 24}$ .
Solve $\boldsymbol{x^2 + 2x - 24 = 0}$ : $\boldsymbol{(x - 4)(x + 6) = 0}$ .
$\boldsymbol{x = 4}$ (positive value).

Answer: Width = 4 m, Length = 6 m.

Sample Questions with Answers

Solve $\boldsymbol{x^2 - 5x + 6 = 0}$ .
Answer: $\boldsymbol{x = 2, x = 3}$ .
Find the nature of roots for $\boldsymbol{x^2 + 4x + 4 = 0}$ .
Answer: $\boldsymbol{\Delta = 0}$ , one repeated root ( $\boldsymbol{x = -2}$ ).
Solve using the quadratic formula: $\boldsymbol{x^2 - 3x - 10 = 0}$ .
Answer: $\boldsymbol{x = 5, x = -2}$ .

FAQs

Where are quadratic equations used?admin2024-12-18T11:08:52+05:30

Introduction to Quadratic Equations

Introduction to Quadratic Equations

What is a Quadratic Equation?

Standard Form of Quadratic Equations

Example:

Methods to Solve Quadratic Equations

1. Factorization

2. Completing the Square

3. Quadratic Formula

4. Graphical Method

Nature of Roots

Applications of Quadratic Equations

Sample Questions with Answers

FAQs

Related Topics

Related Posts

Table of Contents

Join Deeksha Vedantu

> PU + Competitive Exam CoachingPreferred Choice For Toppers25+ Years of Academic Excellence70k+ Success Stories

Related Pages

Latest Posts

Top Links

Top Resources

Top Links

Contact Us

Head Office

Get Social