Complete the Square

Complete the Square

Please take notes**

Last class we learned how to find the roots of a polynomial. From the roots we can find the x intercepts.

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This parabola has roots of p and q so the x intercepts are

(p,0) and (q,0)

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Multiple Choice

What would my x intercepts be for y= 3(x+5)(x + 9)

• (5,0), (9,0)

• (0,15), (0,9)

• (5,9)

• (-5,0), (-9,0)

Multiple Choice

In review of factoring, can you factor x² + 20x + 100

• (x + 10)²

• (x + 20)(x +100)

• (x + 20)²

• (x + 10)(x - 10)

y = (x - h)² + k = (x - h)(x - h) + k

When we multiply this out we get..= x² - 2xh + h² + k

a determines parabola width

(h,k) is the vertex (or round point of the parabola)

So if we have something that looks like

x² + bx + c

x² + bx + (b/2)² - (b/2)² + c

(x - b/2)² -b² + c = (x - h)² + k

Example1: x² + 6x + 4 = a(x-h)² + k

...........................................has vertex (h,k)

b=6, h=b/2=3.............................. a(x² - 2hx +h²) + k

...............................................................x² + 6x + 3² - 3² + 4 = (x + 3)² - 5

...............................................................has vertex (-3,-5)

Ex1: x² + 6x + 4

............= (x + 3)² -5

with vertex (-3,-5)

It has 2 roots or x intercepts but we can't find them by factoring.

Ex2: x² + 2x - 3

........(x² + 2x + 1) -1 -3

......=(x + 1)² - 4 = (x - h)² + k

What is my vertex?

Multiple Choice

Given (x + 1)² - 4 = (x - h)² + k

What are the coordinates of vertex (h,k)?

• (1,-4)

• (-1, 4)

• (-1, -4)

• (1,,4)

What is the vertex (h,k) of

(x + 1)² - 4 = (x - h)² + k

1 = -h, -4 = k

-1 = h, -4 = k, (h,k)=(-1,-4)

Multiple Choice

Ex3: Your turn: x² + 8x +10

...........................(x² + 8x + __² ) - __ + 10

...........................(x + __)² - __

What is my vertex form of the polynomial?

• (x + 8)² + 10

• (x + 8)² +18

• (x + 4)² + 26

• (x + 4)² - 6

(x² + 8x + __² ) - __ + 10

(x² + 8x + 4² ) - 16 + 10

(x + 4)² - 6

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Multiple Choice

Where is my vertex for (x + 4)² - 6?

• (4, -6)

• (4, 6)

• (-4, -6)

• (-4, 6)

(x + 4)² - 6 = (x - h)² + k for vertex (h,k)

4 = -h, -6 = k

-4 = h, -6=k so vertex is (-4, -6)

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Multiple Choice

Ex3: 3x² + 12x + 5 = 3(x² + 4x + __) -3(__) + 5

What is my vertex form?

• (x + 12)² - 7

• (x + 4)² + 1

• 3(x + 4)² + 1

• 3(x + 2)² -7

3x² + 12x + 5 = 3(x² + 4x + __) -3(__) + 5

...........................3(x² + 4x + 2²) - 3(4) + 5

.......................... 3(x + 2)² - 7

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Multiple Choice

Given 3(x + 2)² -7, what is the vertex?

• (2, -7)

• (-2, -7)

• (6, -7)

• (-6, -7)

3(x +2)² - 1 = a(x -h)² + k

2 = -h

-2 = h, -1 = k

So the vertex is (-2, -1)

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We know how to complete the square to get the vertex form and the vertex, now we want to solve for the roots by setting the vertex form = 0.

We may need a calculator for square roots.

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rounded to nearest hundreth x= - .76 or -5.24

Ex1: (x + 3)² -5 has

vertex (-3, -5)

x intercepts (- .76, 0), or ( -5.24,0)

Now we can find roots or x-intercepts that are not whole numbers!

And now we can graph more accurately

Ex4: Find the roots of 3(x + 2)² -9 =0

...........................................3(x+2)² = 9

Divide by 3 ..................... (x + 2)² = 2

Take sq root................... x + 2= \/2

x = -2 - 1.41 or x = -2 + 1.41

x = -3.41 or -.59

Multiple Choice

Your turn to solve Ex2: (x + 1)² - 4 = 0

• 1, -4

• 1, -3

• 1, -2

• -1, 4

Ex 2: Find the roots of (x + 1)² - 4 

0= (x+1)² - 4

4 = (x + 1)² Now take the sq root

2 = x + 1 or -2= x + 1

2-1=x .....or -2-1 = x

My roots are 1 and -3

My x-intercepts are (1,0) and (-3,0)^

Independent practice time

IXLS Geo V14 Find the vertex (up to 70)

.........Alg2 K10 Solve for roots ..(up to 50)

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