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This is an inline equation: $E = mc^2$ and a block equation:

Find $x^2$. % KaTeX inline notation Inline notation: % KaTeX inline notation Inline notation: \(\varphi = \dfrac{1+\sqrt5}{2}= 1.6180339887…\)

Quiestion 1:

This is an inline equation: $E = mc^2$ and a block equation:

Find $x^2$. % KaTeX inline notation Inline notation: % KaTeX inline notation Inline notation: \(\varphi = \dfrac{1+\sqrt5}{2}= 1.6180339887…\)

This shortcode allows you to solve a quadratic equation of the form $ax^2 + bx + c = 0$ by just entering the coefficients.

Example 1: Two Real Roots

This solves the equation $x^2 - 5x + 6 = 0$.

Example 2: One Real Root

This solves the equation $y^2 + 2y + 1 = 0$.

Example 3: Complex Roots

This solves the equation $x^2 + x + 1 = 0$.

Example 4: A Blank Solver

Try your own coefficients below.

This interactive tool uses pure JavaScript to find the vertex, focus, and directrix for any parabola in the form $y = ax^2 + bx + c$.

Example 1: A simple parabola

Solve for the properties of $y = 0.5x^2 + 2x - 3$.

Example 2: Another example

Solve for the properties of $y = -x^2 + 6x - 8$.

Try your own

Enter your own coefficients below.

This tool finds the unique parabola equation ($y = ax^2 + bx + c$) that passes through two given points and has a specific slope at the first point.

Example 1: Finding a Simple Parabola

Let’s find the equation for the parabola passing through (0, 1) and (2, 5) with a slope of 2 at x=0. (The answer should be y = x² + 2x + 1)

Example 2: Another Case

Find the parabola passing through (1, 6) and (3, 2) with a slope of 5 at x=1.

Your Own Points

Enter your own points and slope to find the equation.

This tool finds the unique cubic equation ($y = ax^3 + bx^2 + cx + d$) that passes through two given points and has a specific slope at each of those points.

Example 1: Finding a Simple Cubic

Find the equation for the cubic passing through (0, 1) and (2, 9) with slopes of 0 and 12 at those points. (The answer should be y = x³ + 1)

Example 2: Another Case

Find the cubic passing through (-1, 0) and (1, 4) with slopes of 3 and 15 at those points.

Your Own Points and Slopes

Enter your own values to find the equation.

Let’s find the value of a for the parabola equation $y=ax^2 + bx + c$ that passes through points (3, 2) and (2, 3), with slopes 31 at x=3 and 14 at x=2.

A Consistent Example

Let’s find the parabola for two points and slopes that are consistent. For a parabola with the equation $y = x^2 + 2x + 1$:

• Passes through (0, 1), with slope 2.
• Passes through (2, 9), with slope 6.

  Find Parabola Equation from Two Points and Slopes

  y = ax² + bx + c

  Point 1 (x₁, y₁):

  Slope at x₁ (m₁):

  Point 2 (x₂, y₂):

  Slope at x₂ (m₂):

  Find Coefficients

  Results will appear here.

This tool calculates the shortest distance between two lines in 2D. Enter the coefficients for each line in the form Ax + By + C = 0.

Example 1: Parallel Lines

Calculate the distance between 3x + 4y + 10 = 0 and 6x + 8y + 5 = 0.

Example 2: Intersecting Lines

The distance is 0 for intersecting lines like 2x + y - 5 = 0 and x - y + 1 = 0.

Example 3: Coincident Lines

These are the same line, so the distance is 0.

Your Own Lines

Enter the coefficients for your own lines below.

This tool helps you find the ratio in which a point P divides the line segment joining points A and B.

Example 1: Internal Division

Find the ratio in which point P(3, 4) divides the segment from A(1, 2) to B(6, 7).

Example 2: External Division

Find the ratio in which point P(9, 10) divides the segment from A(3, 4) to B(5, 6).

Example 3: Non-collinear Points

What happens when the points don’t lie on the same line?

Your Own Coordinates

Enter the coordinates of your own points below.

This tool uses the coordinates of a triangle’s vertices to calculate its area.

Example 1: A Right Triangle

Let’s find the area of a simple right triangle with vertices (0,0), (4,0), and (0,3).

Example 2: The Example from the Code

Find the area of a triangle with vertices (0,0), (4,0), and (2,3).

Example 3: Collinear Points

What happens when the points are on the same line?

Your Own Triangle

Enter the coordinates of your own triangle below.

This interactive tool allows you to solve for the value of ‘x’ in a linear equation. Just type your equation in the box below and hit “Solve”.

Example 1: Your Equation

Here is the example you provided, pre-filled.

Example 2: Another Equation

Try solving this one: 3x + 10 = 2x - 5

Example 3: Identity Equation

This one has infinite solutions.

Your Own Equation