Equation Rendering Test

This page tests the rendering of mathematical equations using MathJax in LaTeX format. We use the $$ characters before and after an equation and we clearly list the definitions of the variables after each equation where new variables are introduced.

Linear Equations

Equation 1: Slope-Intercept Form

The slope-intercept form of a linear equation:

\[y = mx + b\]

where:

• \(m\) is the slope
• \(b\) is the y-intercept

Equation 2: Point-Slope Form

The point-slope form of a line passing through point \((x_1, y_1)\) with slope \(m\):

\[y - y_1 = m(x - x_1)\]

Quadratic Equations

Equation 3: Standard Form of a Quadratic

The standard form of a quadratic equation:

\[ax^2 + bx + c = 0\]

where:

• \(a \neq 0\).

Equation 4: Quadratic Formula

The quadratic formula for solving \(ax^2 + bx + c = 0\):

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

Equation 5: Vertex Form

The vertex form of a quadratic function:

\[y = a(x - h)^2 + k\]

where:

• \((h, k)\) is the vertex of the parabola.

Exponents and Radicals

Equation 6: Product Rule for Exponents

When multiplying powers with the same base:

\[a^m \cdot a^n = a^{m+n}\]

Equation 7: Power Rule for Exponents

When raising a power to a power:

\[(a^m)^n = a^{mn}\]

Equation 8: Radical Expression

Converting between radical and exponential form:

\[\sqrt[n]{a^m} = a^{\frac{m}{n}}\]

Systems and Factoring

Equation 9: Difference of Squares

Factoring the difference of two squares:

\[a^2 - b^2 = (a + b)(a - b)\]

Equation 10: System of Linear Equations

A system of two linear equations in two variables:

\[\begin{cases} ax + by = c \\ dx + ey = f \end{cases}\]

Inline Equations Test

You can also use inline equations like \(x^2 + 5x + 6 = 0\) or the discriminant \(\Delta = b^2 - 4ac\) within regular text.

The distance formula \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) is essential for coordinate geometry.

Complex Expressions

Completing the Square

To complete the square for \(x^2 + bx\):

\[x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2\]

Compound Interest Formula

The compound interest formula:

\[A = P\left(1 + \frac{r}{n}\right)^{nt}\]

where: - \(A\) = final amount - \(P\) = principal (initial amount) - \(r\) = annual interest rate (decimal) - \(n\) = number of times interest is compounded per year - \(t\) = time in years

Rendering Verification

If MathJax is configured correctly, all equations above should render beautifully with proper mathematical notation. If you see raw LaTeX code (e.g., $$y = mx + b$$), then MathJax is not loading properly.

Expected behavior: - Block equations centered and displayed on their own lines - Inline equations properly integrated with text - Fractions, square roots, and exponents properly formatted - Greek letters and mathematical symbols rendered correctly