Quadratic Relationships

By: Tavia Martin

What Is Quadratics & How Is It Useful

The name Quadratic derives from "quad" meaning square (x2).It is also called an "Equation of degree 2" (because of the "2" on the x).Some real life examples can be: Mcdonald's logo , rides at wonderland, , a football throw up in the air, and many more. Without the use of Quadratics, a lot of careers would be unavailable, such as Architects who use this unit of the grade 10 math to build amazing buildings, towers and more.


Table Of Contents

Types of equations

- Factored form

- Standard form

- Vertex form

Factored form

- Factors and zeroes

- Expanding (factored form to standard form)

- Common factoring

- Factoring simple trinomials

- Factoring complex trinomials

- Special cases

  • Difference of squares
  • Perfect square

- Factoring by grouping

- Graphing factored form

Vertex form

- Isolating for x

- Graphing vertex form

Standard form

- Solving using quadratic equation

- Discriminant

- Completing the square (standard to vertex)

- Graphing from standard form

Word problems


Factors and Zeros

Factors: A factor is one of the linear expressions of a single-variable polynomial. A polynomial can have several factors, such as the factors...
(x - 1) and (x + 3).

The factors of a polynomial are important to find because they can be multiplied together to gain a polynomial.

When polynomials are graphed, many of them intersect the x-axis. The locations where a polynomial crosses the x-axis are called ‘zeros.’

Zeros: A zero is the location where a polynomial intersects the x-axis. These locations are called zeros because the y-values of these locations are always equal to zero.

When polynomials are graphed, many of them intersect the x-axis. The locations where a polynomial crosses the x-axis are called ‘zeros.’ ,

Expanding (Factored Form to Standard)

Expanded Notation: Writing a number to show the value of each digit. It is shown as a sum of each digit multiplied by its matching place value (units, tens, hundreds, etc.)

The factored form of a quadratic is useful for finding the roots and other properties but sometimes it is useful to expand in order to simplify the equation.

Expanding from factored form uses the distributive property of real numbers which states for real numbers x and y,

c (x + y) = cx + cy

Standard Notation: Just the number as we normally write it. Example: 382 is in Standard Notation.

Common Factoring

Common Factors: Factors are the numbers you multiply together to get another number

When you find the factors of two or more numbers, and then find some factors are the same ("common"), then they are the "common factors".

Example: 12 and 30
• The factors of 12 are: 1, 2, 3, 4, 6 and 12
• The factors of 30 are: 1, 2, 3, 5, 6, 10, 15 and 30
So the common factors of 12 and 30 are: 1, 2, 3 and 6
Common Factoring Tutorial

Factoring Simple Trinomials

Factoring Simple Trinomials

Factoring Complex Trinomials

Factoring Complex Trinomials

Difference of squares

Factoring difference of squares

Perfect Squares

Factoring perfect square trinomials

Factoring By Grouping

Graphing Factored Form

Step 1:Decide if the four terms have anything in common, called the greatest common factor or GCF. If so, factor out the GCF. Do not forget to include the GCF as part of your final answer.

Step 2:Create smaller groups within the problem, usually done by grouping the first two terms together and the last two terms together.

Step 3:Factor out the GCF from each of the two groups. In the second group, you have a choice of factoring out a positive or negative number. To determine whether you should factor out a positive or negative number, you need to look at the signs before the second and fourth terms. If the two signs are the same (both positive or both negative) you need to factor out a positive number. If the two signs are different, you must factor out a negative number.

Step 4: If the factors inside of the parenthesis are exactly the same, it is time for the 2 for 1 special. The one thing that the two groups have in common should be what is in parenthesis, so you can factor out what is inside the parenthesis, but only write what is inside the parenthesis once. If what is inside the parenthesis does not match, you need to rearrange the four terms and try again until you get a perfect match. If you have rearranged the problems a couple of times and still have not found a perfect match, then the problem does not factor.

Step 5: Determine if the remaining factors can be factored any further.


Isolating for 'x'

step by step:

1. Combine like terms.

2. Simplify the equation.

3. Move terms with X's in them to the left side of the equation.

4. Move all parts of the equation except the terms that have an x in them to the the right side of the equation.

5. Divide both sides of the equation by the coefficient.

6. Solve for x.

Graphing Vertex Form

Graphing a parabola in vertex form


Solving Using Quadratic Equation

Solve Quadratic Equations using Quadratic Formula


In algebra, the discriminant of a polynomal is a function of its coefficients, It gives information about the nature of its roots.

Discriminant of Quadratic Equations

Graphing in Standard Form

graph parabola standard form

Word Problems

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  1. · Factored form : a(x-r)(x-s)
  2. · Vertex form: a(x-h)²+k
  3. · Standard form: ax²+bx+c