Activity Set 3.5.iPREP PPTX

Visual Algebra for Teachers

Chapter 3

REAL NUMBERS AND QUADRATIC FUNCTIONS

Visual Algebra for Teachers

Activity Set 3.5i)

Completing the Square, the Quadratic Formula and

Quadratic Graphs

Visual Algebra for Teachers

PURPOSETo learn:How to use squares and square roots while solving quadratic equations. How to complete the square to find the quadratic formula and the y - a(x - h)2 +k form of a quadratic function. How the graphs of general quadratic functions differ from the graph of the simplest parabola: y = x2.

To be able to analyze any quadratic function.

Black and red tiles, white and opposite white n-strips and black and red x-squares

Graphing calculator with table functions (recommended)

MATERIALS

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INTRODUCTION

Squares and Square RootsIn our previous work, we noticed that equations such as

were difficult to solve.

However if we “set everything equal to zero,” this allowed us to use quadratic rectangle arrays to solve such equations.

It turns out, if components of quadratic equations are square rectangular arrays, we can use additional techniques to solve these equations.

2 6 7x x

2 6 7 0x x

Squares and Square RootsSuppose we wish to solve an equation such as:

Previously we would approach this by setting everything equal to zero:

Then factoring:

2 4 0x

2 4x

2 2 0x x

( 2)( 2) 0  if 2 0  or 2 0x x x x

However, because (i) everything on the left side of the equation (x2) is square and

(ii) everything on the right side of the equation is a number

We can use an additional, and in this case, faster technique for determining the solutions to

Squares and Square Roots

2 4x

2 4 0 if 2 or 2x x x

Solution methods for x2 = 4

(as shown on the next slides)We can think of the new technique in terms of:

algebra pieces and

In terms of graphing

Algebra Piece Solution for x2 = 4The dimensions of the black x-square must be

.

2

2 2  or  2 2

If 4, then 2.x x

Graphing Solution for x2 = 4Think of the intersection of two functions.

It is easy to see graphically the two intersection points

are(-2, 4) and (2, 4).

.

2 and 4y x y

Squares and Square Roots is a particularly easy example.

Both the left side and the right side of the equation are already square.

Suppose we wish to solve an equation such as:

In this case we cannot make 2 black tiles into a square array shape. Let’s look at this new equation using algebra pieces and using a graph.

.

2 4x

2 2x

Algebra Piece Solution for x2 = 2

The dimensions of the black x-square are:x x or –x -x

If we can find a number whose square is 2, the opposite of that number should also have a square equal to 2.

By definition, the square root of 2 ( ) is the number whose square is 2.

2

Algebra Piece Solution for x2 = 2Thus, according to the algebra piece model, the dimensions of the black x-square must be

To solve:

we “take the square root of both sides” and keep in mind that we should determine both the positive and the negative answer.

2 2 or 2 2

2 2x

2

2 2 or 2 2

If 2, then 2x x

Graphing Solution for x2 = 2To solve, we can also think of the intersection of two functions.

It is easy to see graphically if x is the x-value of an intersection point of y=x2 and y = 2, then so is –x. This parallels our algebra piece work

.

2 and 2y x y

2If 2, then 2x x

Taking the square rootNotice this new technique “take the square root of both sides” is really all we did for

The two examples we have looked at both have a simple x2 on the one side of the equation.

However, our new technique: Taking the Square Root of Both Sides works if one side is any square and the other side is any positive number (why does the number have to be positive?).

2

Since 4 2,

if 4, then 4 2x x

2 4x

You are now ready for:

PREP QUIZ 3.5.i

See Moodle

Visual Algebra for Teachers