Answer:
b = 1/3 x = -1/7 and -1/2
Step-by-step explanation:
(b−5)x^2−(b−2)x+b=0
Since we know one of the solutions, we can solve for b
(b−5)(.5)^2−(b−2)(.5)+b=0
(b−5)(.25)−(b−2)(.5)+b=0
Distribute
.25b -1.25 -.5b+1 +b = 0
Combine like terms
.75b -.25 =0
Add .25 to each side
.75b = .25
Divide by .75
b = .25/.75
b =1/3
Substituting this back into the equation
(1/3−5)(x)^2−(1/3−2)(x)+1/3=0
Getting a common denominator
(1/3−15/3)(x)^2−(1/3−6/3)(x)+1/3=0
-14/3 x^2 +5/3x +1/3 =0
Multiplying by 3 to get rid of the fractions
-14x^2 +5x +1 =0
using the quadratic formula
-b ±sqrt( b^2-4ac)
-----------------------------
2a
-5 ±sqrt( 5^2-4(-14)1)
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2(-14)
-5 ±sqrt( 81)
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2(-14)
-5 ±9
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-28
-5+9 -5-9
--------- and ---------
-28 -28
-4/28 and 14/28
-1/7 and 1/2
