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Quadratics

Learning Goals:

I can put a standard form quadratic equation into vertex form by completing the square

I can answer word problems involving quadratics

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Quadratics

Quadratics that are perfect squares have “c” values that are equal to 2.

For example: y = x2 + 4x + 4 is a perfect square trinomial because:

b = 4c = 2

= 22

= 4

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Quadratics

Why is y = x2 + 4x + 4 called a perfect square?

Algebra Tiles

x2 x 1

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Quadratics

We can represent y = x2 + 4x + 4 using algebra tiles:

x2x1

xx

x1

11

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QuadraticsPerfect Square Trinomial are always in this form:

a2x2 2abx + b2

And when we factor a perfect square trinomial, it will always factor like this:

(ax

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QuadraticsIs this an example of a perfect square trinomial?

16x2 +8x +2

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QuadraticsFactor this perfect square trinomial:

25x2 - 30x + 9

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QuadraticsRemember that you can factor out any number, even if it leaves you with a rational number.Example: y = 2x2 + 3x + 9

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QuadraticsDetermine the “c” value that would make each of the following perfect squares:

a) y = x2 + 5x + c

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QuadraticsDetermine the “c” value that would make each of the following perfect squares:

c) y = x2 + 9x + c

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Quadratics

Completing the square is the process we go through to change a standard form quadratic equation into vertex form.

What do you notice?Standard Form Vertex Form

x2 + 2x + 3 (x + 1)2 + 2x2 + 4x + 1 (x + 2)2 - 3x2 + 6x + 8 (x + 3)2 – 1x2+ 4x + 5 2(x + 1)2 + 3

3x2 + 18x + 12 3(x + 3)2 - 152x2 + 8x + 7 2(x + 2)2 - 1

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Quadratics

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QuadraticsExample 1: Put into vertex form

Step One: If there is an “a” value, factor it out of the ”x2” and the “x” term.

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QuadraticsExample 1: Put into vertex form

Step Two: Find a constant that must be added inside the brackets to make a perfect square.

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QuadraticsExample 1: Put into vertex form

Step Three: Inside the brackets, add and subtract the constant you found in step 2.

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QuadraticsExample1 : Put into vertex formStep Four: Group the three terms that make the perfect square together. Move the subtracted term outside the brackets by first multiplying it by the common factor.

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QuadraticsExample 1: Put into vertex formStep Five: Factor the perfect square.

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QuadraticsExample 1: Put into vertex form

Step Six: Collect like terms.

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QuadraticsExample 2: Put into vertex form

Step One: If there is an “a” value, factor it out of the ”x2” and the “x” term.

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QuadraticsExample 2: Put into vertex form

Step Two: Find a constant that must be added inside the brackets to make a perfect square.

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Quadratics

Step Three: Inside the brackets, add and subtract the constant you found in step 2.

Example 2: Put into vertex form

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Quadratics

Step Four: Group the three terms that make the perfect square together. Move the subtracted term outside the brackets by first multiplying it by the common factor.

Example 2: Put into vertex form

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Quadratics

Step Five: Factor the perfect square.Example 2: Put into vertex form

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Quadratics

Step Six: Collect like terms.Example 2: Put into vertex form

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QuadraticsExample 3: Put into vertex form

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QuadraticsA football is kicked into the air. Its height, , in metres, after seconds is approximated by the equation: . What is the maximum height reached by the football?

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QuadraticsA computer game company models the profit on its latest game using the equation : , where is the number of games sold in hundred thousands, and is the profit in millions of dollars. What is the maximum profit the company can expect to earn using this model, and how many games do they need to sell to make that profit?

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Quadratics

Homework

Pg. 2 #8-10, 13 Diagnostic Test - Tuesday