Use completing the square to express the solutions to each of these quadratic equations in the form a +/- b√n?
where a + b are rational numbers and n is an integer.
a) x^2-x-3=0 b) x^2-1/2x-1=0
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where a + b are rational numbers and n is an integer.
a) x^2-x-3=0 b) x^2-1/2x-1=0
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a) x² - x - 3 = 0
Add 3 to both sides.
x² - x = 3
Add blanks to represent what's missing.
x² - x + ______ = 3 + ______
Take the coefficient of the middle term: -1
Divide it by 2: (-1 / 2)
Square it: (-1 / 2)² = (-1)² / (2)² = (1 / 4)
Add this to both sides (by filling in the blanks).
x² - x + (1 / 4) = 3 + (1 / 4)
The LHS is now a perfect square.
[x - (1 / 2)]² = 3 + (1 / 4)
Simplify the RHS.
[x - (1 / 2)]² = 3(4 / 4) + (1 / 4)
[x - (1 / 2)]² = (12 / 4) + (1 / 4)
[x - (1 / 2)]² = (13 / 4)
Now you can solve.
[x - (1 / 2)]² = (13 / 4)
Take the square root of both sides.
√[x - (1 / 2)]² = √(13 / 4)
x - (1 / 2) = ± √(13 / 4)
x - (1 / 2) = ± (√13 / √4)
x - (1 / 2) = ± (√13 / 2)
Add (1 / 2) to both sides.
x = (1 / 2) ± (√13 / 2)
Simplify if desired.
x = (1 ± √13) / 2
ANSWER: x = (1 / 2) ± (√13 / 2) (equivalent to x = (1 ± √13) / 2)
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b) x² - (1/2)x - 1 = 0
Add 1 to both sides.
x² - (1/2)x = 1
Add blanks to represent what's missing.
x² - (1/2)x + ______ = 1 + ______
Take the coefficient of the middle term: (-1 / 2)
Divide it by 2: (-1 / 2) / 2 = (-1 / 4)
Square it: (-1 / 4)² = (-1)² / (4)² = (1 / 16)
Add this to both sides (by filling in the blanks).
x² - (1/2)x + (1 / 16) = 1 + (1 / 16)
The LHS is now a perfect square.
[x - (1/4)]² = 1 + (1 / 16)
Simplify the RHS.
[x - (1/4)]² = 1(16 / 16) + (1 / 16)
[x - (1/4)]² = (16 / 16) + (1 / 16)
[x - (1/4)]² = (17 / 16)
Now you can solve.
[x - (1/4)]² = (17 / 16)
Take the square root of both sides.
√[x - (1/4)]² = √(17 / 16)
x - (1/4) = ± √(17 / 16)
x - (1/4) = ± (√17 / √16)
x - (1/4) = ± (√17 / 4)
Add (1 / 4) to both sides.
x = (1/4) ± (√17 / 4)
Simplify if needed.
x = (1 ± √17) / 4
ANSWER: x = (1/4) ± (√17 / 4) (equivalent to x = (1 ± √17) / 4)
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Question a)
x = [ 1 ± â (1 + 12 ) ] / 2
x = [ 1 ± â (13) ] / 2
Question b)
x = [ (1/2) ± â (1/4 + 4 ) ] / 2
x = [ (1/2) ± â (5/4) ] / 2
x = [ (1/2) ± (1/2)â5 ] / 2
x = 1/4 ± (1/4)â5
x = (1/4) ( 1 ± â5 )
this answer can be done dy using formula
-b+/-âb^-4ac /2a
-1+/- â1+12 /2*2
= -1+/-â13 /4
= -1+/-3.605 /4
x= -0.9875 or x= -1.90125
in the same same war try the other