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Quadratic Formula in LaTeX: Complete Guide

November 5, 202510 min read

Introduction

The quadratic formula is one of the most fundamental formulas in algebra, providing a method to solve any quadratic equation of the form ax² + bx + c = 0. This formula is essential for students, educators, and professionals working with polynomial equations.

In LaTeX, typesetting the quadratic formula requires proper use of fractions, square roots, and mathematical notation. This complete guide covers everything you need to write the quadratic formula correctly in LaTeX, from the basic formula to variations and practical examples. This comprehensive guide provides step-by-step instructions for mastering the quadratic formula in LaTeX.

The Standard Quadratic Formula

The quadratic formula solves for the roots of any quadratic equation ax² + bx + c = 0 where a ≠ 0. The formula provides both solutions:

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

This is the most common form of the quadratic formula. The ± symbol indicates that there are typically two solutions: one using addition and one using subtraction.

LaTeX Code for Standard Form

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

Understanding the Formula Components

Let's break down each part of the quadratic formula:

a, b, c: The coefficients of the quadratic equation ax² + bx + c = 0
±: The plus-minus symbol, indicating two possible solutions
b² - 4ac: The discriminant, which determines the nature of the roots
√(b² - 4ac): The square root of the discriminant
2a: The denominator, twice the coefficient of x²

The Discriminant

The discriminant Δ = b² - 4ac determines the nature of the roots:

Δ = b^{2} - 4 a c

If Δ > 0: Two distinct real roots
If Δ = 0: One repeated real root (a double root)
If Δ < 0: Two complex conjugate roots

Alternative Forms of the Quadratic Formula

Separated Solutions

Sometimes it's clearer to write the two solutions separately:

x_{1} = \frac{- b + b ^{2} - 4 a c}{2 a}, x_{2} = \frac{- b - b ^{2} - 4 a c}{2 a}

x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a}, \quad x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a}

Using the Discriminant

When the discriminant is defined separately, you can write:

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

Vertex Form Connection

The quadratic formula can also be expressed in terms of the vertex:

x = h \pm \frac{k}{a}, where (h, k) is the vertex

Practical Examples

Example 1: Simple Quadratic

Solve x² - 5x + 6 = 0

Here, a = 1, b = -5, and c = 6:

x = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( 1 ) ( 6 )}{2 ( 1 )} = \frac{5 \pm 25 - 24}{2} = \frac{5 \pm 1}{2}

This gives us x = 3 or x = 2.

Example 2: With Fractional Coefficients

Solve 2x² + 3x - 1 = 0

x = \frac{- 3 \pm 3 ^{2} - 4 ( 2 ) ( - 1 )}{2 ( 2 )} = \frac{- 3 \pm 9 + 8}{4} = \frac{- 3 \pm 17}{4}

Example 3: Complex Roots

Solve x² + 2x + 5 = 0

x = \frac{- 2 \pm 2 ^{2} - 4 ( 1 ) ( 5 )}{2 ( 1 )} = \frac{- 2 \pm 4 - 20}{2} = \frac{- 2 \pm - 16}{2} = \frac{- 2 \pm 4 i}{2} = - 1 \pm 2 i

Since the discriminant is negative, we have two complex conjugate roots.

LaTeX Typesetting Tips

Proper Spacing

Use \quad or \qquad for spacing between multiple formulas on the same line.

Parentheses and Brackets

When using negative numbers in the formula, ensure proper parentheses:

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

Display Mode vs Inline

Use display mode (centered, larger) for standalone formulas, and inline mode for formulas within text:

Display mode: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Inline mode: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Common Variations and Special Cases

When b = 0

For equations of the form ax² + c = 0:

x = \pm - \frac{c}{a}

When c = 0

For equations of the form ax² + bx = 0:

x = 0 or x = - \frac{b}{a}

Monic Quadratic (a = 1)

When the leading coefficient is 1, the formula simplifies slightly:

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

Conclusion

The quadratic formula is a fundamental tool in algebra, and knowing how to typeset it correctly in LaTeX is essential for academic and professional mathematical writing. Whether you're writing a research paper, creating educational materials, or preparing a presentation, proper LaTeX formatting ensures your formulas are clear and professional.

Remember to use appropriate spacing, parentheses for clarity, and choose between display and inline modes based on your document's needs. With practice, typesetting the quadratic formula and other mathematical expressions in LaTeX becomes second nature.