M3 Completing the Square and Factoring

M3 Completing the Square and Factoring

Completing the Square Step 1

Rearrange the equation by moving the c value to the other side of the = sign.

Move the c value 9 to the other side of the equal sign.

Multiple Choice

What is the first step to solving THIS equation by completing the square?
a² + 10a + 21 = 0

• Set the equation equal to zero
• Divide 10 by 2 and add the result to both sides
• Add a² and 10a together
• Subtract the 21

Multiple Choice

To solve by completing the square, what needs to be moved in this equation?
x² = 9 - 4x

• the -4x
• the 9
• the x²

Multiple Choice

What is the first step to solve the equation by completing the square

x²+ 4x - 24 = 0?

• Subtract 24 from both sides

• Add 24 to both sides

• Divide b by 2 and square it

• Add b/2 to both sides

Multiple Choice

 $x^2-5=-14x$  
When completing the square, what must be done first? 

• Move 14x to be with  $x^2$  by adding 14x to both sides
  

• Move -5 to the other side by addition.
  

• Move 14x to be with  $x^2$  by adding 14x to both sides AND move -5 to the other side by addition.

• Nothing, the first step has already been done.  

Completing the square:  Step 2

 $+\left(\frac{b}{2}\right)^2$  

 $x^2+6x\ \ \ =16$  

+ $\left(\frac{b}{2}\right)^2$  

ADD  $\left(\frac{b}{2}\right)^2$  to both sides

 $\left(\frac{6}{2}\right)^2=9$  

Multiple Choice

What do you do to the b value to correctly complete the square?

• square it
• divide it by 2 and square the result
• divide it by 2 and take the square root of the result
• divide it by 2 only

Multiple Choice

Given the equation  $x^2+18x=0$  how do you find the number to add to both sides to help you complete the square?  

• Divide 18 by 2, then square that number and add it to both sides of the equation

• Square 18 and then add it both sides of the equation

• It already is a perfect square

• Subtract 18x from both sides and then divide both sides by x to find the answer

Multiple Choice

 $x^2+20x-3=0$  
When completing the square, what number goes in the empty space?
 $\left(x+10\right)^2=3+\ldots$  

•  $10$  

•  $20$  

•  $100$  

•  $3$  

Multiple Choice

 $x^2-5=-14x$  
When completing the square, what number goes in the empty space?
 $\left(x+7\right)^2=5+\ldots$  

•  $7$  

•  $14$  

•  $49$  

•  $-14$  

Multiple Choice

Find the value of "c" that completes the square. Then rewrite the trinomial as a perfect square.

 $x^2-2x+c$  

•  $1;\ \left(x-2\right)^2$  

•  $4;\ \left(x-2\right)^2$  

•  $1;\ \left(x-1\right)^2$  

•  $4;\ \left(x-1\right)^2$  

Multiple Choice

Find the value of "c" that completes the square. Then rewrite the trinomial as a perfect square.

 $n^2-20n+c$  

•  $10;\ \left(n-10\right)^2$  

•  $100;\ \left(n-10\right)^2$  

•  $100;\ \left(n+10\right)^2$  

•  $10;\ \left(n+10\right)^2$  

Multiple Choice

Find the value of "c" that completes the square.

 $t^2+7t+c$  

•  $\frac{7}{2}$  

• 3.5

•  $\frac{49}{4}$  

•  $\frac{49}{2}$  

Multiple Choice

What did you need to add to both sides to complete the square?

  $x^2-12x+_{ }$  ___ =  20 + ___

• 6

• 36

• 10

• 144

Step 3:   Factor the perfect square trinomial

•  $ax^2+bx+c=\left(x+\frac{b}{2}\right)^2$  

•  $x^2-6x+9,\$  b = -6

•  $\frac{b}{2}=\left(\frac{-6}{2}\right)=-3$  

•  $\left(x+\frac{b}{2}\right)^2$  = $\left(x-3\right)^2$  

Multiple Choice

 $x^2+10x=-3$  
When completing the square, what number goes in the empty space?
 $\left(x+\ldots\right)^2=22$  

•  $5$  

•  $10$  

•  $25$  

•  $-15$  

Multiple Choice

 $x^2+8x=5$  
When completing the square, what number goes in the empty space?
 $\left(x+\ldots\right)^2=21$  

•  $4$  

•  $8$  

•  $16$  

•  $3$  

Multiple Choice

When factoring

x² - 4x + 4 = 20,

what goes in the blank?

(x - __ )² = 20

• 4

• 2

• 8

• 20

Multiple Choice

Complete the Square
x² + 6x = 5

• (x + 3)² = 5
• (x + 6)² = 9
• (x + 3)² = 14
• (x + 6)² = 14

Factoring Trinomials and Zero Product Property

•  $ax^2+bx+c\ =0\ \rightarrow\left(x\pm\right)\left(x\pm\right)$  

•  $ab=0,\ a\ =\ 0\ ,\ b\ =\ 0$  

Multiple Choice

In the equation
 (x + 2)(x - 3) = 0, the binomials are called...?

• Factors
• Products
• Multiples
• Zeros

Multiple Choice

Factor
n² + 16n + 63

• (n-7)(n+4)
• (n+7)(n-9)
• (n-3)(n-10)
• (n+7)(n+9)

Multiple Choice

Factor
k²+7x+10

• (k+2)(k+5)
• (k-2)(k-5)
• (k+10)(k+4)
• (k+2)(k-5)

Multiple Choice

Factor Completely:  15t² - 27t - 6

• 3(5t +1)(t - 2)
• (3t - 6)(5t + 1)
• 3(5t - 1)(t + 2)
• (15t + 6)(1t - 6)

Multiple Choice

Factor Completely:  3n² - 15n + 18

• (n - 2)(n - 3)
• 3(n + 2)(n + 3)
• (3n - 9)(n - 2)
• 3(n - 3)(n - 2)

Multiple Choice

Factor
4n²-17n-15

• (n-10)(6n+1)
• (4n-5)(n-3)
• (n-5)(4n-3)
• (n-5)(4n+3)

Multiple Choice

Factor
3v² - 4v - 7

• (3v-7)(v+1)
• 3(v-7)(v-1)
• (3v+1)(v-9)
• (3v+1)(v-10)

Multiple Choice

What is the first step in solving 
(x + 2)(x - 3) = 0      ?

• Plug in zero for x
• Solve for x
• FOIL
• Set each binomial factor equal to zero

Multiple Choice

Rewrite in factored form and identify the solution.

 $x^2-6x+9=0$  

•  $(x-3)(x-3)=0;\ x=3$  

•  $(x+3)(x+3)=0;\ \ \ x=-3$  

•  $(x+4)(x+5)=0;\ \ \ x=-4\ and\ x=-5$  

• Cannot be factored; no real solution

Multiple Choice

Solve and find your x-intercepts:
0=x(x - 6)

• x=0 and x=-6
• x=1 and x=-6
• x=0 and x=6
• x=6

Multiple Choice

Solve for the values of x:
(x + 2)(x - 3) = 0

• { 2, 3 }
• { -2, -3 }
• { 2, 3 }
• { -2, 3 }

Multiple Choice

 Solve for the values of x.

 $2x^2+7x-4$ 

• { 1/2, -4 }

•  { 1/2, 4 }

• { 2, -4 }

• { -2, 4 }