# MPM2D

## Classify and Solve Quadratic Equations (Part 2)

As mentioned in the previous lesson, trinomial quadratics containing a first degree and second degree term, along with a constant, are called complete quadratics. Just like pure and incomplete quadratics, solving complete quadratics is as simple as setting y = 0 and solving for x. Remember, to solve an equation means to find the values of…

## Apply Slope, Midpoint, and Length Formulas

This section shows you how to apply geometry and algebra to a variety of problems. Many of these problems involve several steps that require different skills. Developing a problem solving process is particularly important for dealing with such problems. These four steps can help you solve any multi-step problems: 1. Understand…

## Right Bisector of a Two Lines

In this section, we will learn how to create an equation that represents the right bisector. A right bisector is a line that passes through the midpoint of a line at 90 degrees; it is sometimes called a perpendicular bisector. Given that it is a line, all lines can be represented in form: y…

## Median of a Triangle

A median of a triangle is a line segment (shown in red) joining any vertex (corner) to the midpoint of the opposing side, bisecting it. We learned in the previous lesson what the midpoint means for a line. Every triangle has exactly three medians, one from each vertex, and they all intersect each other…

## Translate Statements Into Algebraic Expressions

To change sentences into mathematical expressions and equations, look for important words like these: Common Identifier Translates to Increased, Sum, More $+$ (plus) Decreased, difference, less $–$ (minus) Twice, doubled $\times 2$ Tripled $\times 3$ The same $=$

## Solve Problems Involving Right Triangles

Many application problems involving right triangles mention the angle of depression and angle of elevation. The angle of depression is angle measured below the horizontal, and may also be called the angle of declination. The angle of elevation is the angle measured above the horizontal (see figure below), and may also be called the angle of inclination. Let’s…

## The Tangent, Sine, and Cosine Ratio

To start, a ratio is a mathematical comparison. In other words, the comparison of any two quantities is called a ratio. Any time you work with fractions, for example, you’re technically comparing the numerator quantity to the denominator: null The Tangent Ratio If you have a right triangle, and you…

## Use Similar Triangles to Solve Problems

Similar triangles are formed when you have two different triangles both sharing the same three angles. This makes their lengths, while different, proportional to one another. This suggests that similar triangles are not congruent figures, meaning that they’re not identical in both length and angles, as shown below.The first video below…

## Solve Problems Using Quadratic Equations

So far in this unit, you have learned a variety of methods for solving quadratic equations: graphing, factoring, completing the square, and the quadratic formula. In this final section, you will learn how to apply these skills to solve problems related to situations that can be modelled by quadratic relations,…

## The Quadratic Formula

Of the several methods we have for solving quadratics, the most useful is the quadratic formula. It will work for any quadratic, regardless of the type of roots, and it can easily be programmed into your calculator. Interestingly, the quadratic formula is derived by completing the square of the general form…

## Graph Quadratics Using the x-Intercepts

Arguably, the whole purpose behind learning how to factor quadratics and find its roots (also known as zeros) is to know how to sketch parabolas accurately. As you have come realize, many quadratic relations of the form y = ax² + bx + c can be factored to find the…

## Classify and Solve Quadratic Equations (Part 1)

To solve a quadratic equation means to find values for x that make the y side of the equation equal to zero. Recall that a quadratic has a highest degree of 2. Generally, the highest degree in any equation dictates the maximum possible number of solutions. Thus a quadratic equation, being…

## Determining the Maximum and Minimum of Quadratics

All quadratic equations have at least one minimum or maximum value. Think of a parabola, they either open upwards like a smile ∪ or face downwards like a frown ∩. What dictates the direction of opening is the a term. If the a term is positive, it open upwards (a minimum). If the…

## Quadratic Relation in Factored Form

Unlike the vertex form of a quadratic (y = a(x – h)² + k) which exposes the vertex of a parabola (h, k), the factored form (y = a(x – r)(x – s)) exposes the roots of the parabola – that is, if they exist. For example, if we have a…

## Graph Vertex Form Quadratic

One thing to be mindful of quadratics is that they come in many different forms. Take for example, y = 2x² + 2x – 4. A quadratic whose x² and x term are visible is in its general form (in bold for clarity). This equation can be rewritten in two other…

## Transformations of Quadratics

The simplest quadratic equation is: y = x² If you were to graph this using a table of values, it would look like the graph on the left. Notice that vertex is directly at (0, 0). But a world where all parabolas are fixated at the origin would be boring.…

## Quadratic Relations

A polynomial equation of second degree (i.e. x²) is called a quadratic equation. It is common practice to refer to it simply as a quadratic. A quadratic is in general form when it is written in the following form, where a, b, and c are constants: y = ax² +…

## Properties of Circles

This last lesson in this unit will demonstrate a few important properties of circles. When you first start learning about circles, you learn what radius and diameter mean. As a refresher, the diameter is the measurement across the circle passing through the center (shown below), while the radius is half that…

## Verify Properties of Quadrilaterals

A quadrilateral is a polygon (i.e. 2D shape) with four edges (or sides) and four vertices or corners. Examples of quadrilaterals are illustrated below. In this section we’ll focus on parallelograms and trapezoids. The videos below will prove the following three things: The diagonals of a parallelogram bisect each other. Joining the…

## Properties of Triangles

Before we start looking at examples, let’s outline some important properties of triangles. The medians of a triangle meet at a single point, the centroid. The centroid is a fancy word for an objects center of gravity (see the animation below). Each median bisects (splits into two equal parts) the area…