Grade 10 Quadratics

BY: Karman Gakhal

Table of contents

Introduction picture

1. Expanding and Simplifying

  • Factoring Standard Form
  • Common Factoring
  • Factoring by Grouping
  • Simple Trinomials
  • Complex Trinomials
  • Difference of Squares
  • Perfect Squares
  • Completing the Square

2. Graphing

-Standard Form

  • x Intercepts
  • Axis of Symmetry
-Vertex Form
  • Each Part of Equation
  • Types of Graphs
  • Transformations
  • How to Graphs
-Factored Form
  • x Intercepts
  • Axis of Symmetry
  • Optimal Value

3. Standard form

  • Learning Goals
  • Summary
  • Quadratic Formula
  • Completing the Square
  • Word Problem

4. Solving

  • Solving Standard Form
  • Solving Vertex Form
  • Solving Factored Form

5. Word Problems

6. Reflection

7. Connections

8. Useful links

What is a parabola?

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Solving: Expand and Simplify

Factoring Standard Form

Now that you have learned how to form standard form from factored form, you now need to learn how to factor standard form. You have been learning how to factor from earlier grades but only with two numbers. Now you are going to learn how factor with numbers and variables. Factoring is not as complicated as it sounds. It is explained step by step in the video below through the guess and trial method.

Learning Goals

How to common factor

how to factor by grouping

How to factor simple trinomials

How to factor complex trinomials

How to find the difference of squares

How to find factor perfect squares

Basic formula for factoring


The value of a gives you the shape and direction of opening

The value of r and s give you the x-intercepts

Solve using the factors

Types of Factoring:

Greatest Common Factor

Simple factoring (a=1)

Complex factoring

Special case - Difference of squares

Special case – Perfect square

Common Factoring

You can tell if it a factor is common because there well only be 2 terms and 1 variable. The way you would solve it is by finding the numbers that both terms have in common and and divide the expression by it putting the remaining terms in a bracket and the number in common outside of it.


Factoring by Grouping

You have probably learned about the distributive property from earlier grades and that is exactly what you need to know to expand and simplify factored form. A factored expression looks like this: (x+5)(x+6). One of the main thing you need to know is that when you expand and simplify an expression like this is that you need to multiply each number, variable and both by each number or variable in the second bracket.

Simple Trinomials

This is a very simple method, and you get the your answer in factor by grouping question is like the opposite of factor by grouping. The point of it is to get 2 Binomials that multiply to get the given trinomial. It is a simple trinomial if there is 3 being x² the other being x with no power and it can have a number in in font, and the last being a number with no variable.( note: there does not have to a x term or a term with no variable.)


Complexed Trinomials

Complexed Trinomials are very similar to the simple trinomial it follows the same x²+bx+c formula but it adds an "a" in front of x². to find the answer you would do the same-thing but there would be more possibilities. Also when you get your two brackets the X variable well have a number in front of it when well multiply with other terms inside the other bracket. The video well help to further explain.


Deference of Squares

Deference of squares start of with 2 brackets like factoring by grouping. But the thing is the terms inside both brackets are the same. But the signs are different someone well have to be positive and one would have to be negative.E.g (x+2) (x-2). The way you would solve it is also the same as factoring by grouping.


Perfect Squares

Perfect squares are like deference of squares with a little change to it. Both expressions have to have the same terms in both brackets but unlike deference of squares the signs stay the same. In other words the the brackets are the exact same.


Completing the Square

Completing the square is the method we use to form vertex form from standard form. What you do is: divide 'b' by 2 and square root that number. Add that number into the equation but since we are not allowed to change the equation, subtract that number from the equation too. Now that you have that new number, it will act as your 'c'. Ignore the subtracted number and your original 'c'. Then factor the equation you have made which will be a perfect square trinomial. Then simplify the factored form. Now go back to the two numbers that you had put aside and collect the like terms. In the end it should look like a vertex form equation.

Word Problem

The Height of a ball thrown can be approximated by the formula h=-2t^2+4t+48, where t is the time in seconds and h is the height in metres.

1)Write the formula in factored form


2)When will the ball hit the ground

6 seconds (t=6)

3)What is the maximum height the ball will reach in metres?



Standard Form

To graph, you need to find the x intercepts through quadratic formula. Then you need to find the vertex through axis of symmetry. You cannot find the axis of symmetry the same way you find it in factored form. For this there is a special formula. To find the axis of symmetry you need to divide negative 'b' by 2a. Then substitute the axis of symmetry into the standard form equation to get 'y'. The axis of symmetry and 'y' are the vertex (x, y).
Solving Quadratic Equations by Graphing

Vertex Form

For most people Vertex form is the easiest form to graph because most of the information in in the equation. The structure of the vertex form is y = a(x-h)²+k.

Learning Goals

1. How to solve for the xbow and y intercepts

2. How to graph vertex form

3. Equation for vertex form

4. Variables in vertex form

5. What a standard graph looks like

6. How to make a standard graph match your parabola

7. First and Second difference

8. Step Pattern Method

Each Part of Equation

A= tells us the stretch(+) /compression (-) of the graph
H= tells us how many units left/right the vertex is going to move from 0
K= tells us how many units up/down the vertex is going to move from 0
X= the the 'y' are coordinates on the graph


Math has its own language and there is specific terminology that you must know. When explaining the transformations of certain coordinates of a vertex form graph, there are certain words we use. These words are:

  • Vertical Stretch: If the 'a' is a number greater than 1 then the graph would have a vertical stretch which means that is steeper. The way you would write it: This graph has been vertically stretched by the factor of 2 (or any number greater than 1).
  • Vertical Compression: If the 'a' is a number less than 1 then the graph would have a vertical compression meaning it would be wider. The way you would write it: This graph is vertically compressed by the factor of 0.5 (or any number less that 1).
  • Vertical Reflection: If the 'a' is a negative number then it means the graph is vertically reflected over the x axis downwards.
  • Translations: If there is a number after x squared like in y=a(x-h)²+5 (the 'h') then it means that the vertex has been translated right or left. The way you would write it: The vertex has been horizontally been translated 3 (or any other number) units to the right/left. Also if there is a 'k' then that means that the vertex has been translated up or down. The way you would write it: The graph has been vertically translated 4 (or any other number) units up/down.
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Quick Way of Graphing a Quadratic Function in Vertex Form

Step patterns (Basic)

Over 1 -- Up 1

Over 2 -- Up 4

Types of Graphs

How to Graphs

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Solving Standard Form

To solve standard form we do not use the same method we used for factored form. This one is a little more complicated. To solve a standard form equation we use the quadratic formula. It would be a good idea to memorize the formula. You can look at the images below to learn how to solve standard form through quadratic formula.

Solving Vertex Form

To solve a vertex form equation we use the method of isolation. You need to substitute 'y' as '0' and then begin to isolate 'x'. This will give you your two x intercepts. To solve for the y intercept, you must substitute 'x' as '0'. If you need a more thorough explanation then you can look at the step by step instructions below.

Factored Form

Once you have learned how to solve factored form it is easy to graph. To graph factored form you would need to find the two x intercepts through solving by substituting 'y' as 0. Then you would need to find the axis of symmetry to find the vertex. To find the axis of symmetry you would need to add the two x intercepts and divide that number by two. The axis of symmetry will be the x of you vertex (x, y). Then to find the 'y' sub in the x as the axis of symmetry into the factored form equation. The answer you get will be the 'y' of the vertex. The vertex is the optimal value- the lowest or highest coordinate on the graph. Now you will have the vertex and the two x intercepts and you can easily graph the factored form equation.

Standard Form

Learning Goals

  1. What is the quadratic formula used for
  2. what is the quadratic formula
  3. how to complete squares


The quadratic formula is a formula that was created in order to find the the x-intercepts of a given parabola. In order to use the quadratic formula your equations must be in the form: y= ax^2 + bx + c. Completing the square is another way to find the vertex and is when you get the equation listed above and convert it into vertex form: y= a(x-h)^2 + k. This equation then allows you to find the vertex which is (-h,k) = (x,y)

Quadratic Formula

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Completing the Square

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3.14 Completing the square

Word Problems


A ball is thrown upwards at an initial velocity of 8.4m/s, from a height of 1.2 m above the ground. The height of the ball, in meters, above the ground after t seconds is modeled by the equation h=-4.9t²+8.4t +1.2.

1. How long will it take for the ball to fall to the ground, rounded to the nearest tenth of a second?

2.What is the maximum height of the ball? At what time will it reach this height? Round your answers to the nearest tenth.

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Factor Quadratic Expressions

The flight of a ball is modeled by the equation ℎ = −5² + 20 + 25 where h represents the height of the ball in meters, and t represents the amount of time the ball has spent in the air in seconds.

a. Write the equation in factored form, determine the x-intercepts and vertex, and graph.

b. When does the ball hit the ground?

c. What is the highest the ball flies, and when does that happen?

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Overall, I have enjoyed quadratics very much. I found this unit a bit harder compared to other units we have already done. The few specific concepts I have enjoyed are: factoring, distributive property, quadratic formula. The reason I enjoy these concepts is because I like to do algebra and found these concepts easier then the rest. Furthermore, I like to use different methods and formula's to solve equations. Although I enjoyed this unit, there were a few things that I did not like which includes graphing and word problems. I did not like graphing because I do not like to apply my knowledge and put the information into a graph. I can easily graph, it is not difficult for me but it is not something I enjoyed as much because of the amount of time it takes. I also did not like the word problem concept because I found it somewhat difficult. The reason I found word problems difficult was because I did not know what formula or method to use to solve the word problem. I know how to do all the methods but I do not understand when to use them in a word problem. Once I understand what I need to use to solve a word problem the it becomes easy to solve. This is the main reason I do not enjoy word problems. This is why I want to improve on the word problem concept. I will improve by practicing and continue to solve word problems. By practicing I will be able to recognize when I need to use a certain method to solve. I have also done a lot of work that I am proud of in math class, specifically the quadratic unit. The best work I have done is the quadratics 2 unit test part 2 which i believe was the application part . I have done very well on this test because I was very comfortable with the different types of forms and I feel it is the most simple form of a quadratic units. In the end, this was a fun and educational unit and I have learned a lot of information that I thoroughly understand and enjoy.


In quadratics 1, we mostly focused on solving vertex form by finding axis of symmetry, optimal value, zeros, and determining the step pattern. With all this information, we were able to graph it.

In quadratics 2, we focused on expanding and factoring. We learnt how to find axis of symmetry, optimal value, and zeros. We also did this in quadratics 1, but the only difference was that there might have been some different methods to do this.

In quadratics 3, we focused on rewriting standard form equations into vertex form by completing the square and by learning a new equation which is called the Quadratic Formula. We also learnt how to graph quadratics using the x-intercepts which we also learnt in quadratics 1.

In quadratics 1, 2 & 3, the word problems were mostly all asking for the same stuff just in different forms of equations and different methods of asking them. For example, what is the max height? They are asking for you to find the vertex and state the 'y'. In vertex form {y=a(x-h)²+k}, the vertex is (h,k) and so you already know your max height by just looking at the equation and knowing 'k' is your 'y-intercept'. In standard form {y=ax²+bx+c}, you know that the 'c' is the 'y-intercept' which is your max height. Now in factored form, you have to do some solving to find the max height, starting with finding the zeros, then the AOS, & lastly the optimal value. The AOS and optimal value would be your vertex, at which the optimal value is the 'y-intercept'.

As you can see, all of the parts of quadratics are related in several ways.

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