square_root_property.doc by Ron Mower 9/23/02 9:40 AM Page 1 of 2
Square Root Property
The Square Root Property says:
You can take the square root of both sides of an equation if you put a plus/minus on the right side.
As a result, two equations are generated…the Plus version, and the Minus version…you get two answers.
Of course, there would be no sense in doing this unless the variable term is squared or is trapped in a squared term.
When the x is alone and squared (Form 1)
ü
ü
x2 = c
x=± c
x = c or x = − c
ü
When the x is part of a (linear) binomial and the binomial is squared (x is trapped) (Form 2)
ü
( x + a) 2 = c
ü
x+a = ± c
x = − a + c or x = − a − c
ü
when the x is alone
1) x = 4 (Form 1)
ü The x-term is missing
ü x can be plus or minus 2 (can be written as x = ±2 or as, x = 2 or x = −2 )
2
ü General form is x = c (solutions can be written as x = ± c or as, x =
ü Example
2
c or x = − c )
x2 = 5
x = ± 5 (take the square foot of both sides - when taking an even root, always put a plus/minus in front - on the right side only)
x = 5 or x = − 5
(we now write the two answers separately - not always necessary)
−
2) ( x + 4) = 9 (Form 2)
ü The square is of a binomial
ü x + 4 can be plus or minus the square root of -9
2
ü General form is ( x + h) = k
ü Example
2
( x + 4) 2 = − 9
x+4=±
−
9
(take the square foot of both sides - when taking an even root, always put a plus/minus in front - on the right side only)
x = −4 ± 3 − 1 = −4 ± 3i
x = −4 + 3i or x = −4 − 3i
3)
ü
ü
ü
(subtract 4 from both sides - always put the number in front of the plus/minus sign)
(we now write the two answers separately - not always necessary)
x 2 + 5 = 9 (Variation of Form 1…put in Form 1)
The x-term is missing
The quadratic is not in one of the above forms
Put in Form 1 by subtracting 5. For example:
x2 + 5 = 9
x2 + 5 − 5 = 9 − 5
x2 = 4
x = ±2
x = 2 or x = −2
(subtract 5 from both sides…or add -5 to both sides)
(compute)
(take the square foot of both sides - when taking an even root, always put a plus/minus in front - on the right side only)
(we now write the two answers separately - not always necessary)
Page 1 of 2 \\RMHP\D_HP1\My Documents 2000\Word\Math\xReference\Alg_Ref\square_root_property.doc System HP1
square_root_property.doc by Ron Mower 9/23/02 9:40 AM Page 2 of 2
4)
ü
ü
ü
( x + 5) 2 − 10 = 18 (Variation of Form 2…put in Form 2)
The square is a binomial
The quadratic is not in one of the above forms
Put in Form 2 by adding 10. For example:
( x + 5) 2 − 10 = 18
( x + 5) 2 = 10 + 18
( x + 5) 2 = 28
(add 10 to both sides)
(compute)
x + 5 = ± 28
(take the square foot of both sides - when taking an even root, always put a plus/minus in front - on the right side only)
x = −5 ± 28
(subtract 5 from both sides…or add -5 to both sides - always put the number in front of the plus/minus sign)
x = −5 + 28 or x = −5 − 28 (we now write the two answers separately - not always necessary) Please see the Note below
Note: Since
28 = 7(4) = 2 7 , the above answer can be written as x = −5 + 2 7 or x = −5 − 2 7
5) A Quickie (Form 1)
ü
ü
ü
x2 = 0
x = ±0
x=0
(since the zero repeats, ie. +0 = 0 and -0 = 0, there are two answers but they are both the same…we say, "zero of multiplicity 2")
6) A Quickie (Put in Form 1)
ü
ü
ü
ü
x 2 −1 = 0
x2 = 1
x = ±1
x = 1 or x = −1
end
Page 2 of 2 \\RMHP\D_HP1\My Documents 2000\Word\Math\xReference\Alg_Ref\square_root_property.doc System HP1