Methods of Solving Quadratic Equations

Methods of Solving Quadratic Equations

Goal: I can use the most efficient method to solve quadratic equations

Poll

Today, we are going to focus on solving quadratics using the three methods below. Which one do you feel most confident using?

• Square Roots

• Factoring

• Quadratic Formula

Poll

Which method do you feel least confident using?

• Square Roots

• Factoring

• Quadratic Formula

Reviewing Methods of Solving

We will use the next slides to refresh ourselves on each method. Then, we will determine which method is BEST to use.

Taking Square Roots

•  $ax^{2\ }+\ c\ =\ 0$  or  $ax^{2\ }=\ c$  The equation is missing its "bx" term.

• Isolate the  $x^{2\ }$  term

• Take the square root of both sides

• Don't forget the +/- 

Square Root Example

Multiple Choice

Solve by finding square roots and simplify completely:
 $x^2-16=0$ 

•  $x=\pm16$  

•  $x=4$  

•  $x=\pm4$  

• No real solutions

Multiple Choice

Solve by finding square roots and simplify completely:

9k² = 225

• x = 4

• 
  

  $x=\pm4$

• $x=\pm25$

• $x=\pm5$

Factoring

•  $ax^{2\ }+bx\ +c=0$  

• Must be in standard form and equal zero

• If c term is missing ( $ax^2+bx$  ) , factor GCF

• After factoring, set both factors equal to zero and solve for x. 

Factoring Using GCF Example

Multiple Choice

Solve by factoring the GCF

 $10x^2-8x=0$  

•  $x=2;\ x=-\frac{5}{4}$  
  

•  $x=2;\ x=\frac{5}{4}$  
  

•  $x=0\ ;\ x=\frac{4}{5}$  

•  $x=0;\ x=-\frac{4}{5}$  

Factoring Trinomial Example

Multiple Choice

Solve by factoring:
 $x^2+10x+24=0$ 

•  $x=6\ or\ x=4$  

•  $x=\pm\sqrt{14}$  

•  $x=-6\ or\ x=-4$  

• No real solutions

Quadratic Formula

• Must be in standard form and set equal to zero

• Used when the equation is NOT factorable

• ALL equations can be solved using the quadratic formula

Quadratic Formula Example

Multiple Choice

Use the quadratic formula to find the solutions for

2x² - 9x + 5 = 0

• x = -8.14 and x= 6.14

• x = -4.07 and x= 3.07

• x = 3.85 and x = 0.65

• ZERO solutions

Which method should I use?

Certain equations can be solved using multiple methods. Others can only be solved one way.

Multiple Choice

Solve x^2 - 3x - 4 = 0 using any method.

• x = 4 and -1

• x = -4 and -1

• x = 4 and 1

• x = -4 and 1

Poll

Which method did you use to solve x^2 - 3x - 4 = 0 ?

• Square Roots

• Factoring

• Quadratic Formula

Multiple Choice

Solve using any method

16a² - 9 = 0

• a = ^-3/₄, ³/₄

• a = ^-4/₃, ⁴/₃

• a = ^-9/₁₆, ⁹/₁₆

• a = ^-16/₉, ¹⁶/₉

Poll

Which method did you use to solve 16a² - 9 = 0 ?

• Square Roots

• Factoring

• Quadratic Formula

Let's Determine the Most Efficient Methods

All these methods are great, but we need to know which ones work best for each equation.

Multiple Choice

Decide on the best method:

2x² - 7x + 2 = 0

• Square Root Method

• Factor Method

• Quadratic Formula

Multiple Choice

Decide on the best method:

x² - 3x - 54 = 0

• Square Root Method

• Factor Method

• Quadratic Formula

Multiple Choice

Decide on the best method:

4x² - 25 = 0

• Square Root Method

• Factor Method

• Quadratic Formula

Solve the following questions using the most efficient method.

Multiple Choice

Solve using the method of your choice

• x= -2, x= 4

• x= -8

• x =±8

• x = -2.03 , x = 8.33

Multiple Choice

Solve using the method of your choice.

• x= ±81

• x= ±9

• x= 4, x = 9

• No solution

Multiple Choice

Solve using the method of your choice: x² - 8x = 20

• x = 10 and x = -2

• x = -10 and x = -2

• x = 10 and x = 2

Multiple Choice

Solve by the method of your choice.

• x= 4, x =-1

• x=8/3, x= 9/5

• x= 1/5, x= 3/7

• x= 7, x= -2

Multiple Choice

Solve using the method of your choice: x² + 13x + 12 = 0

• x = -12, -1

• x = 12.03, 1.55

• x = 13, 1

• x = -13, -1

Multiple Choice

Solve using the method of your choice

2x² + 2x − 12 = 0

• x = −2.67, 3.86

• x = 2.67, 3.86

• x = 2, −3

• x = −2, −3

Multiple Choice

 $x^2-7x=0$  

Solve by the method of your choice.

• x = 7 only
  

• x = 0 only
  

• x = 0 and 7
  

• There is no real answer for this equation.
  

Multiple Choice

Solve using the method of your choice

9x² = 4 + 7x

• x = 2 and x = 3

• x = -1 and x = .25

• x = 1 and x = -.45

• x = 1.16, and x = -.38

Multiple Choice

Solve by the method of your choice

• x = 5 , x = 7

• x=2/3, x=6

• x=4/7, x=6

• x=-3/8, x = -7

Poll

How confident are you in your ability to solve quadratic equations using the method of your choice?

• Not confident. I need more help.

• Somewhat confident. I can do most but I struggle with a few.

• Confident. I can do this on my own,

• Very Confident. I could teach this to someone.