[The quadratic formula][section 67]
(1.) x2 - 4x = 21
-21 -21
subtract 21 from each side
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x2 - 4x - 21 = 0
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x = [-b ± √b2 -
4ac]/(2a) use the quadratic formula
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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
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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
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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
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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
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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
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x = [-b ± √b2 - 4ac]/(2a) use the quadratic formula
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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
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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)