Multiplication Laws in Equations

Multiplication is one of the 4 basic operations of arithmetic along with addition, subtraction and division. In a multiplication question, the quantities that are multiplied are called factors and the answer is a product. Multiplication is the one operation that can be represented using many different symbols.

You can indicate multiplication with:

•  ‘X’, usually with numbers (e.g. 5 x 12)
•   ‘*’, an asterisk, is used in spreadsheets such as Excel (e.g. 5 * 12 or 5*C2)
•  ‘⋅’ , an interpunct, is used with numbers and variables (e.g. 5⋅12 nor a⋅b)
• ‘  ’ , no symbol, is used with variables or variables and numbers. In this case, the numbers are called “coefficients”. (e.g. ab or 5c)
• ‘( )’ ,brackets or parentheses, are used with only numbers, only variables or a combination of numbers and variables (e.g. (5)(12) = 60; (x²)(x) = x³; (2x)(3x)= 6x²)

When multiplication is the operation used in algebra, it is the last three that are used most frequently. The choice of symbol depends on the writer and what the writer is describing.

When you first learned to multiply, you probably learned some facts about multiplication with numbers. You have also learned that the same facts apply when you are working with variables. The facts were recorded a long time ago and are often referred to as laws.  It is not necessary to know the names of the laws but rather that you can use the laws when doing mathematics.

 The laws of arithmetic and algebra for multiplication include:

• commutative law, which states that the order of multiplying terms will not change the answer. i.e. ab = ba or 3 x 7 = 7 x 3,
• identity law, which states that multiplication of any number by 1 results in the number. i.e. a⋅1 = a or 1 x 9 = 9,
• multiplication by zero, which states that anything multiplied by 0 equals 0 i.e. 0 x a = 0 or 5 x 0 = 0,
• zero product law, which is another way to view the previous statement and is frequently used with algebra,
  • If the product of two quantities is 0, then at least one of the quantities must be 0, i.e. If ab = 0 then either a = 0, or b = 0, or both a and b are 0.

Watch the following video for more about the zero product law:

Solve the following equations. Think about the above laws that lead you quickly to the solution.

1. 5x = 0
2. 3(x - 2) = 0
3. -4(x + 3) = 0
4. 7(x - 8) = 0

 

Practice

Determine values for x that would make the equation: x(x + 7) = 0 true.

Answer

Forms of the Equation for Quadratic Relations

When you looked more closely at linear relations, you learned that different equations can be written to describe the same linear relation. Each form provides certain information about the linear relation. The information is usually related to key features of linear relations.

In Unit 2, you identified the key features of parabolas and learned the vocabulary that is associated with the key features. Like linear relations, there are different forms of a quadratic equation that will produce the same graph, and each form can provide information about the key features of the quadratic relation.  Recall from Unit 2, that the shape of a quadratic relation is a parabola.

 

Let’s investigate two different equations for quadratic relations and identify the information that is available from the equations.

The standard form of the equation for a quadratic relation is y = ax2 + bx + c, the sum of three unlike terms. The values, a, b, and c, determine the shape and position of the parabola. In all the examples that you will be using, the value for ‘a’ will always be 1 (i.e. y = x2 + bx + c).

The factored form of the equation for a quadratic relation is y = a(x - r)(x - s), a product of three factors. The values, a, r, and s, are values that also determine the shape and position of the parabola. In all the examples, the value for ‘a’ will always be 1 (i.e. y = (x - r)(x - s)).

In the examples that follow, integer values will be used for b, c, r, and s to more easily see the connections between the equation and the graph.

You may wish to refer to the Frayer model for Quadratics that you created in Unit 2, Activity 2 for information about the key features that you identified.

Investigating Quadratic Relations

Open the desmos file created for this investigation, and the document Investigating Quadratic Relations.

Save your completed document to your workspace.

In the previous examples, all quadratic relations could be described using both a standard form (y = x2 + bx + c) and a factored form (y = (x - r)(x - s)). You were able to connect key features of the quadratic relation with some of the values in the equations.

Different Forms of Quadratic Equations - Different Parabolas

You probably recall from Unit 2 that all parabolas have a y-intercept, but all parabolas do not necessarily have 2 x-intercepts. Some parabolas have only one x-intercept, while others do not cross the x-axis at all, so have no x-intercepts. The form of the equations for the quadratic relation will change in these situations, but connections between the graph and the form of the equation are still possible. In the following investigation, you will work with some variations of the forms of equations of quadratic relations and make connections between the equation and the graph.

Investigating Different Forms of the Equation

Open the desmos link for Part 2 and the document Investigating Quadratic Relations. Complete this investigation by following the instructions on the document.

Save your completed document to your workspace.

Summary of Connections Between Graphs and Equations

During your investigations, you may have noticed some of the following connections between the graph and the equations. You may have noticed others that are not listed here.

• All quadratic relations can be written in the standard form, y = x2 + bx +  c; the y-intercept is the constant term, ‘c’, in the standard form of the equation.
• The factored form, y = (x - r)(x - s), can only be written when the quadratic relation crosses or touches the x-axis.
• The x-intercepts are the values that will make one of the factors (x - r) or (x - s) equal to 0.
• There can be 2 x-intercepts, 1 x-intercept or no x-intercepts on the graph of a quadratic relation.
• When there are 2 x-intercepts, both can be positive values, both can be negative values, or there can be one positive value and one negative value.
• When there are x-intercepts, the axis of symmetry is in the middle of the two x-intercepts.
• The equation for the axis of symmetry is the equation of a vertical line, x =  (value) and the line passes through the vertex.
• When an x-intercept is 0, the y-intercept is also 0.
• When only one x-intercept is 0, the standard form of the equation has 2 terms.
• When an x-intercept is 0, one factor in the equation is ‘x’. 
• When the vertex is on the y-axis but NOT at (0, 0), the standard form of the equation (y = x2 + bx +  c) has an x2 term and a constant term (meaning that c is not equal to zero), but no ‘x’ term.
• When the vertex is on the y-axis and there are x-intercepts, the values for the x-intercepts are opposite values.
• When the vertex is on the y-axis, the equation of the axis of symmetry is x = 0.
• When the vertex is on the x-axis, the factored form of the equation, y = (x - r)(x - s), has two identical factors (meaning that r = s).

X-intercepts, Factored Form, and the Zero Product Law

You noticed during the investigation that the x-intercepts can be determined from the factored form of the equation, y = (x - r)(x - s). You also know that the zeros are found along the x-axis and the y-coordinate for the ordered pair at the intercept is 0. Let’s look at the connection between the equation, the zeros and the zero product law that you explored during the Minds On.

Using the factored form of the equation, y = (x - 1)(x + 3), you identified that the x-intercepts are located at 1 and -3 (opposite values of -1 and +3). You also graphed the relation and noticed that the parabola crossed the x-axis at two points, (1, 0) and (-3, 0).

At the x-intercept, you know that y = 0 so you can show this in the equation by substituting 0 in for y.

Substituting the value of 0 in for the y-variable. 

y = (x - 1)(x + 3)

0 = (x - 1)(x + 3)  or (x - 1)(x + 3) = 0

This is like the work you did in the Minds On section of this activity.

(x - 1)(x + 3) = 0

Recall the zero product law where you know that at least one of the factors must be equal to 0.

Consider the 2 factors:

(x - 1)(x + 3) = 0

x - 1                                             x + 3

See that either factor can be 0.

x – 1 = 0                                       x + 3 = 0

Show the value for x to make the statement true.

x = 1,  the opposite of – 1                x = -3, the opposite of 3

This equation is true if the value for x is either 1 or -3; there are two possible values which is where the parabola crosses the x-axis.

Practice

Use your knowledge of the zero product law to determine the x-intercepts for the quadratic relation.

y = (x + 5)(x - 1)

Substituting the value of 0 in for the y-variable. 

y = (x + 5)(x - 1)            (x + 5)(x - 1)= 0

Recall the zero product law where you know that at least one of the factors must be equal to 0.

Consider the 2 factors:

(x + 5)(x - 1) = 0

x + 5         x - 1

Show the value for x to make the statement true.

x = -5, the opposite of +5     x = 1, the opposite of -1

This equation is true if the value for x is either -5 or 1; there are two possible values which is where the parabola crosses the x-axis.

Practice

If a quadratic relation has the x-intercepts -3 and 4, which of the following is the correct equation? Give reasons for your choice.
a. y = (x - 3)(x - 4)
b. y = (x + 3)(x + 4)
c. y = (x + 3)(x – 4)
d. y = (x – 3)(x + 4)

Answer

If a quadratic relation has the x-intercepts, -5 and -2, which of the following is the correct equation? Give reasons for your choice.
a. y = (x - 2)(x - 5)
b. y = (x + 2)(x + 5)
c. y = (x + 2)(x – 5)
d. y = (x – 2)(x + 5)

Answer

Equations and Graphs

Select equations from the list that represent the standard form or the factored form of the quadratic relation. Also, include equations that represent the equation of the axis of symmetry for the relation.

Some graphs may have more than one equation. Not all the equations will match a graph.

MatchingEqns

Long Description

You used graphing software to compare two different forms of the equation of a quadratic relation. You found information about the graph from each form and used the appropriate vocabulary to describe the information. You recognized the connection between the values of the x-intercepts and the factored form. You can locate the axis of symmetry from the zeros and determine the equation of the axis.

Congratulations, you have completed Unit 3, Activity 4. You may move on to Unit 3, Activity 5.