[The quadratic formula][section 67]

(1.) x2 - 4x = 21

-21 -21 subtract 21 from each side
________________
x2 - 4x - 21 = 0
________
x = [-b √b2 - 4ac]/(2a) use the quadratic formula
____________
x = [4 √42 - 4*1*-21]/(2*1) make substitutions

x = [4 10]/2 multiply add take square root

x = 7 x = -3 add subtract divide

results: x = 7 , x = -3

(2.) x2 + 6x = 16 here is the problem

-16 -16 subtract 16 from each side
________________
x2 + 6x - 16 = 0 subtract
________
x = [-b √b2 - 4ac]/(2a) use the quadratic formula
____________
x = [-6 √62 - 4*1*-16]/(2*1) make substitutions

x = [-6 10]/2 multiply add and take sq root

x = 2 x = -8 add subtract divide

results: x = 2 , x = -8

(3.) 2x2 - x - 3 = 0 here is the problem
________
x = [-b √b2 - 4ac]/(2a) use the quadratic formula
___________
x = [1 √12 - 4*2*-3]/(2*2) make substitutions

x = (1 5)/4 multiply add take square root

x = 3/2 x = -1 add subtract and divide

results: x = 3/2 , x = -1

(4.) a2 + 4a = 5 here is the problem

-5 -5 subtract 5 from each side
________________
a2 + 4a - 5 = 0 subtract
________
a = [-b √b2 - 4ac]/(2a) use the quadratic formula
___________
a = [-4 √42 - 4*1*-5]/(2*1) make substitutions

a = [-4 6]/2 multiply add & take sq root

a = 1 a = -5 add subtract and divide

results: a = 1 , a = -5

(5.) 6a2 + 13a + 6 = 0 here is the problem
________
a = [-b
√b2 - 4ac]/(2a) use the quadratic formula
___________
a = [-13 √132 - 4*6*6]/(2*6) make substitutions

a = [-13 5]/12 multiply subtract take sq root

a = -2/3 a = -3/2 add subtract reduce

results: a = -2/3 , a = -3/2


(6.) 2y2 = 5y - 3

+ 3 + 3 add 3 to each side
__________________
2y2 + 3 = 5y add

-5y -5y subtract 5y from each side
________________
2y2 - 5y + 3 = 0 subtract
________
y = [-b √b2 - 4ac]/(2a) use the quadratic formula
__________
y = [5 √52 - 4*2*3]/(2*2) make substitutions

y = (5 1)/4 multiply subtract and take sq root

y = 3/2 y = 1 add subtract and reduce and divide

results: y = 3/2 , y = 1


(7.) x2 + 2x = 1 here is the problem

-1 -1 subtract 1 from each side
_____________
x2 + 2x - 1 = 0 subtract
________
x = [-b
√b2 - 4ac]/(2a) use the quadratic formula
___________
x = [-1 √22 - 4*1*-1]/(2*1) make substitutions
_
x = (-1 √8)/2 multiply and add
_ _
x = (-1 √4√2)/2 factor like this
_
x = (-1 2√2)/2 take the square root of the 4
_
x = (-1/2) √2 divide thru by 2, cancel
_ _
results: x = (-1/2) + √2 , x = (-1/2) - √2


(8.) x2 + 6x + 3 = 0 here is the problem
________
x = [-b
√b2 - 4ac]/(2a) use the quadratic formula
__________
x = [-6 √62 - 4*1*3]/(2*1) make substitutions
__
x = (-6 √12)/2 multiply and subtract
_ _
x = (-6 √4√3)/2 factor
_
x = (-6 2√3)/2 take sq root of the 4
_
x = -3 √3 divide thru by 2
_ _
results: x = -3 + √3 , x = -3 - √3


(9.) 2x2 + 7x + 2 = 0 here is the problem
________
x = [-b
√b2 - 4ac]/(2a) use the quadratic formula
__________
x = [-7 √72 - 4*2*2]/(2*2) make substitutions
__
x = (-7 √33)/4 multiply subtract and take sq root
__ __
results: x = (-7 + √33)/4 , x = (-7 - √33)/4


(10.) 6x2 - 3x - 4 = 0 here is the problem
________
x = [-b
√b2 - 4ac]/(2a) use the quadratic formula
___________
x = [3 √32 - 4*6*-4]/(2*6) make substitutions
__
x = (3 √96)/12 multiply add
__ _
x = (3 √16√6)/12 factor
_
x = (3 4√6)/12 take sq root of the 16
_
x = (1/4) (√6/3) separate the fraction and reduce
_ _
results: x = (1/4) + (√6/3) , x = (1/4) - (√6/3)