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Algebra

Quadratic Formula

Solving second-degree polynomial equations

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Study tools

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Source & review

Difficulty

Beginner

Read time

5 min

Prerequisites

Basic Algebra

Review statusNeeds review

Source

FormuLab initial formula library

Initial content draft pending verification against authoritative course or textbook sources.

Definition

Understanding the core concept

The quadratic formula provides a systematic way to find the roots (solutions) of any quadratic equation ax² + bx + c = 0. The ± symbol means there are typically two solutions. The discriminant (b² - 4ac) tells us about the nature of solutions: positive means two real solutions, zero means one solution, and negative means two complex solutions.

Variables & Units

Understanding each component

Symbol	Meaning	Units
$a$	Coefficient of x²	$d im e n s i o n l ess$
$b$	Coefficient of x	$d im e n s i o n l ess$
$c$	Constant term	$d im e n s i o n l ess$
$x$	Unknown variable (solution)	$d im e n s i o n l ess$

Real-World Applications

Where this formula is used in practice

Projectile Motion

Calculate time and position of objects moving under gravity

Optimization Problems

Find maximum or minimum values in business and engineering

Circuit Analysis

Determine resonant frequencies in electrical circuits

Area Calculations

Solve for dimensions when area constraints are given

Worked Example

Step-by-step calculation with real numbers

Problem

Solve the equation 2x² + 5x - 3 = 0 using the quadratic formula.

Given

a = 2

b = 5

c = - 3

Solution

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{- 5 \pm 5 ^{2} - 4 ( 2 ) ( - 3 )}{2 ( 2 )}

x = \frac{- 5 \pm 25 + 24}{4}

x = \frac{- 5 \pm 49}{4}

x = \frac{- 5 \pm 7}{4}

x_{1} = \frac{- 5 + 7}{4} = \frac{2}{4} = 0.5

x_{2} = \frac{- 5 - 7}{4} = \frac{- 12}{4} = - 3

Final Answer

x = 0.5 or x = - 3

Formula Guide

Info

DifficultyBeginner

Read Time5 min

PrerequisitesBasic Algebra

Tools

Source & review

StatusNeeds review

Last reviewedPending

Source

FormuLab initial formula library

Initial content draft pending verification against authoritative course or textbook sources.

Assumptions

• Use SI units unless the formula or course context states otherwise.
• Check notation, sign conventions, and applicability against your course material.
• Confirm variables and assumptions before using the formula in graded or professional work.