Ways to solve a Quadratic

Factoring

100

x² - 8x = 0

x(x-8)=0

x-8=0

x=0, x = 8

200

x² + 2x - 8

(x - 2) (x + 4)

300

2x² - 6 = x

2x^2 - x - 6 = 0

2x^2  + 3x - 4x - 6 = 0

x(2x + 3) - 2(2x+3) = 0

(x-2)(2x+3) = 0

x - 2 = 0  2x+3 = 0

x= 2.    x= -(3/2)

400

2x² - 9x - 35 = 0

(2x+5) (x-7)

500

X³ - 25x = 25 - x²

x³ + x² - 25x - 25 = 0 

x² (x+1) - 25(x+1) = 0

(x² - 25)(x+1) = 0

(x+5)(x-5)(x+1) = 0

x+5 = 0     x -5 = 0.     x+1 = 0

x=-5 x=5   x=-1

Taking Square Roots

100

(X² / 2) + 3 = 11

(X² / 2) + 3 = 11

x² / 2 = 8

X² = 16

x= +/- 4

200

2(x+3)² = 40

(x+3)^2 = 20.     Divide both sides by 2

X+3= +/- √20.     square root both sides

x + 3 = +/- 2 √5.      Break down sqrt 20

x= -3 +/- 2 √5       subtract 3 from boths ides

300

(x-2)² / (-2)   = 2

(x-2)² = -4     multiplying both sides by -2

x-2 = +/- √-4.       square root both sides

no solution since you cant have a negative number in a square root

400

5 = (x-2 / 3)²

+/- √5 = (x-2) / 3   square root both sides

+/- 3 √5 = x-2     multiply both sides by 3

x = 2 +/- 3 √5      

500

(x² + 4)² - 5 = 20 

(x² + 4)² = 25  * add 5 to each side 

x² + 4 = 5 * take the square root of each side 

x² = 1 *subtract 4 from each side 

x = +-1 * take the square root of each side 

Completing the square

100

X² + 14x = 15

(14/2)² = 49

x² + 14x + 49 = 15 + 49

(x+ 7)² = 15 + 49

(x+7)² = 64

x+7 = +/- 8

x = -7 +/- 8

x = 1 

x = -15

200

x²-8x = 9 

x²- 8x +(-4)²=9 + 16 We get -4 and 16 from (-8/2) 

(x-4)²=25

x-4= +/- 5 

x= 4+/- 5

x= 9 x= -1

300

2x²+10 = -16x

2x²+16x = -10

2x²+16x + _____ = -10 + ____

x² + 8x + (4)² = -5 + 16

(x+4)² = 11

x+4 = +/- √11

x= -4 +/- √11

400

X² - 5x = 8

(-5/2)² = 25/4

x² - 5x + 25/4 = 8 + 25/4

(x - 5/2)² = 8 + 25/4

(x-5/2)² = 57/4

x - 5/2 = +/- sqrt (57/4)

x - 5/2 = +/- (sqrt 57) / 2

x = 5/2 +/- (sqrt 57) / 2

500

x²-4x = -2

(-4 x 1/2)² = 4

x²-4x + 4 = -2 + 4

(x-2)² = -2 + 4

(x-2)² = 2

x-2 = +/- √2

x = 2 +/- √2

Quadratic Formula

100

2x² + 2x - 12 = 0

2x² + 2x - 12 = 0

a=2 b=2 c=-12

x=(-b +/- √(b²-4ac)) / 2a

x= (-2 +/- √(2² - 4(2)(-12))) / 2(2)

x= (-2 +/- √100) / 4

x = (-2 +/- 10) /4

x= 2 or x=-3

200

9x² - 6x + 1 = 0

a=9  b=-6  c=1

x= (-b+/- √(b²- 4ac))/ 2a

x= (6 +/- √(36 - 4(9)(1))) / 2(9)

x= (6 +/- √0)/18

x= 6/18

x= 1/3.  <----- simplified 

300

5x²+1 = -3x

5x² + 3x + 1 = 0        subract 3x from both sides

a =5 b=3 c =1.      identify each value

x= (-b+/- √(b²- 4ac))/ 2a

x= (-3+/- √((3)² - 4(5)(1))) / 2(5)

x =( -3 +/- √-11 ) / 10

No real solution

 

     

400

-3x² = 4-6x

 -3x² + 6x - 4 = 0

a=-3 b=6 c=-4

x= (-b+/- √(b²- 4ac))/ 2a

x= (3 +/- √(36-4(-3)(-4))) / 2(-3)

x = (3 +/- √-12) / -6

 No real solution

500

5x² + 2 = 4x + 4 

5x² - 4x -2

a=5  b=-4 b=-2

x= (-b+/- √(b²- 4ac))/ 2a

x = (4+/- √(16 - 4(5)(-2))) / 2(5)

 x = (4 +/- √56) / 10

x = (4 +/- 2 √14) /10

 x = (4/10)  +/-  (2√14)/10

x= (2/5) +/- √(14)/5