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