Completing the Square with Parabolas

Perfect Square Trinomials

Examples x2 + 6x + 9 x2 - 10x + 25 x2 + 12x + 36

Creating a Perfect Square Trinomial

In the following perfect square trinomial, the constant term is missing. X2 + 14x + ____

Find the constant term by squaring half the coefficient of the linear term.

(14/2)2 X2 + 14x + 49

Perfect Square Trinomials

Create perfect square trinomials.

x2 + 20x + ___ x2 - 4x + ___ x2 + 5x + ___

100425/4

Completing the Square with Parabolas

Example 1: Put the following parabola in vertex form bycompleting the square:

Step 1: Move the constant, c, to the other side.

𝑦+20=𝑥2+8𝑥

Completing the Square with Parabolas

Step 2: Find

𝑏2=

82=4

𝑦+20=𝑥2+8𝑥

Completing the Square with Parabolas

Step 3: Take 𝑏

2=82=4

Completing the Square with Parabolas

Step 4: Add to both

sides of the equation.

+16 +16

Completing the Square with Parabolas

Step 5: Factor the perfect square trinomial.

Completing the Square with Parabolas

Step 6: Move the

constant back to the right side.

−36 −36

Can you identify the vertex of this

parabola?

Completing the Square with Parabolas

Example 2: Put the following parabola in vertex form bycompleting the square:

Step 1: Move the constant, c, to the other side.

𝑦−1=𝑥2−4 𝑥

Completing the Square with Parabolas

Step 2: Find

𝑏2=

−42 =−2

𝑦−1=𝑥2−4 𝑥

Completing the Square with Parabolas

Step 3: Take 𝑏

2=−42 =−2 (−2)2=4

Completing the Square with Parabolas

Step 4: Add to both

sides of the equation.

+4 +4

Completing the Square with Parabolas

Step 5: Factor the perfect square trinomial.

Completing the Square with Parabolas

Step 6: Move the

constant back to the right side.

−3 −3

Can you identify the vertex of this

parabola?

Completing the Square with Parabolas

Example 3: Put the following parabola in vertex form bycompleting the square:

Step 1: Move the constant, c, to the other side.

𝑥−25=𝑦2+12 𝑦

Completing the Square with Parabolas

Step 2: Find

𝑏2=

122 =6

𝑥−25=𝑦2+12 𝑦

Completing the Square with Parabolas

Step 3: Take 𝑏

2=122 =6

Completing the Square with Parabolas

Step 4: Add to both

sides of the equation.

+36 +36

Completing the Square with Parabolas

Step 5: Factor the perfect square trinomial.

Completing the Square with Parabolas

Step 6: Move the

constant back to the right side.

−11 −11

Can you identify the vertex of this

parabola?

Completing the Square with Parabolas

Example 4: Put the following parabola in vertex form bycompleting the square:

Step 1: Move the constant, c, to the other side.

𝑦=4 𝑥2−24 𝑥−5

𝑦+5=4 𝑥2−24 𝑥

+5 +5

Completing the Square with Parabolas

Step 2: Find **In order to complete

the square, .**Before we can find ,

we must factor out the . 𝑏

2=−62 =−3

Completing the Square with Parabolas

Step 3: Take 𝑏

2=−62 =−3

Completing the Square with Parabolas

Step 4: Add to both

sides of the equation.

+36 +9¿

Completing the Square with Parabolas

Step 5: Factor the perfect square trinomial.

Completing the Square with Parabolas

Step 6: Move the

constant back to the right side.

−41 −41

Can you identify the vertex of this

parabola?

Completing the Square with Parabolas

You Try!Put the following

parabola in vertex form bycompleting the square:Can you identify the vertex of this

parabola?