Algebra

Quadratic Formula

Solution to quadratic equations ax² + bx + c = 0

About Quadratic Formula

The Quadratic Formula represents solution to quadratic equations ax² + bx + c = 0. This algebra formula is fundamental to mathematical analysis and serves as a cornerstone concept that students and professionals encounter throughout their mathematical journey. Its importance extends beyond pure mathematics into applied fields where quantitative analysis is required.

This formula is essential in Algebra and Pre-calculus. It serves as a building block for more advanced mathematical theory and provides the foundation needed to understand complex mathematical relationships. Whether you're studying mathematics, physics, engineering, or economics, familiarity with this formula enhances your analytical capabilities.

Practical applications of the Quadratic Formula include Physics projectile motion, Engineering optimization, Economics profit maximization, among others. Understanding and correctly applying this formula enables problem-solvers to approach challenges more systematically and efficiently. Mastery of this concept not only expands your mathematical knowledge but also improves your overall quantitative reasoning skills.

Visual Preview

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

LaTeX Code

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

Formula Information

Difficulty Level

Beginner

Prerequisites

Basic algebraSquare rootsArithmetic operationsUnderstanding of equationsOrder of operations

Discovered

9th century

Discoverer

Muhammad ibn Musa al-Khwarizmi

Real-World Applications

Physics projectile motion

Engineering optimization

Economics profit maximization

Geometry parabola problems

Computer graphics

Signal processing

Ballistics calculations

Architectural design

Financial modeling

Control systems

Robotics path planning

Examples

x^{2} + 5 x + 6 = 0 \Rightarrow x = \frac{- 5 \pm 25 - 24}{2} = \frac{- 5 \pm 1}{2} \Rightarrow x = - 2 or x = - 3

2 x^{2} - 8 x + 3 = 0 \Rightarrow x = \frac{8 \pm 64 - 24}{4} = \frac{8 \pm 40}{4} = \frac{8 \pm 2 10}{4} = 2 \pm \frac{10}{2}

x^{2} - 4 x + 4 = 0 \Rightarrow x = \frac{4 \pm 16 - 16}{2} = \frac{4}{2} = 2 (repeated root)

x^{2} + 2 x + 5 = 0 \Rightarrow x = \frac{- 2 \pm 4 - 20}{2} = \frac{- 2 \pm - 16}{2} = - 1 \pm 2 i (complex roots)

Mathematical Fields

AlgebraPre-calculusAnalytical geometryNumber theory

Keywords

quadratic formulaquadratic equationrootsdiscriminantalgebrapolynomialsolving equationsmath formulaquadratic functionparabolavertexaxis of symmetry

Important Notes

The discriminant (b² - 4ac) determines the nature of roots: positive (two distinct real roots), zero (one repeated real root), negative (two complex conjugate roots). The vertex of the parabola is at x = -b/(2a). Always check that a ≠ 0 before applying the formula.

Alternative Names

Quadratic equation formulaABC formulaShridharacharya formulaSridharacharya's formula

Common Usage

Solving quadratic equations

Finding roots of polynomials

Optimization problems

Graph analysis

Finding intersection points

Maximum/minimum problems

Formula Variations

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

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

x_{1}, x_{2} = \frac{- b \pm Δ}{2 a} where Δ = b^{2} - 4 a c

Alternative form: x = \frac{2 c}{- b \mp b ^{2} - 4 a c}

Frequently Asked Questions

What if the discriminant is negative?

When the discriminant (b² - 4ac) is negative, the quadratic equation has two complex conjugate roots. The square root of a negative number involves the imaginary unit i, where i² = -1. For example, if the discriminant is -16, then √(-16) = 4i, resulting in complex roots.

What happens when the discriminant equals zero?

When the discriminant equals zero, the quadratic equation has exactly one real root (also called a repeated or double root). This occurs when the parabola touches the x-axis at a single point (the vertex). In this case, both solutions from the formula are identical.

Can I use the quadratic formula if a = 0?

No, the quadratic formula requires a ≠ 0. If a = 0, the equation is not quadratic but linear (bx + c = 0), which can be solved using simple algebra: x = -c/b (provided b ≠ 0).

What is the difference between the ± symbol?

The ± symbol means 'plus or minus' and indicates that there are two solutions: one using the plus sign and one using the minus sign. For example, if you get x = 2 ± 3, this means x = 2 + 3 = 5 or x = 2 - 3 = -1.

How do I know if I should use factoring or the quadratic formula?

Use factoring when the quadratic expression can be easily factored into two binomials, especially when the coefficients are small integers. Use the quadratic formula when factoring is difficult, when you need exact solutions, or when working with decimals or fractions. The quadratic formula always works for any quadratic equation.

What does the discriminant tell me about the graph?

The discriminant reveals information about the parabola's intersection with the x-axis: positive discriminant means two x-intercepts, zero discriminant means one x-intercept (touching the axis), and negative discriminant means no real x-intercepts (the parabola is entirely above or below the x-axis).

Actions

Open in Workspace

Quick Details

Category: Algebra
Difficulty: Beginner
Discovered: 9th century
Discoverer: Muhammad ibn Musa al-Khwarizmi
Formula ID: quad-formula

Fields

AlgebraPre-calculusAnalytical geometryNumber theory

Keywords

quadratic formulaquadratic equationrootsdiscriminantalgebrapolynomialsolving equationsmath formulaquadratic functionparabolavertexaxis of symmetry

Quadratic Formula

About Quadratic Formula

Visual Preview

LaTeX Code

Formula Information

Difficulty Level

Prerequisites

Discovered

Discoverer

Real-World Applications

Examples

Mathematical Fields

Keywords

Related Topics

Important Notes

Alternative Names

Common Usage

Formula Variations

Frequently Asked Questions

What if the discriminant is negative?

What happens when the discriminant equals zero?

Can I use the quadratic formula if a = 0?

What is the difference between the ± symbol?

How do I know if I should use factoring or the quadratic formula?

What does the discriminant tell me about the graph?

Related Formulas

Binomial Theorem

Difference of Squares

Completing the Square

Actions

Quick Details

Fields

Keywords