(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
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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)