Question 16536: if (a+2)x^2- 2ax - a = 0
what is range of a ?
Answer by AnlytcPhil(1807)   (Show Source):
You can put this solution on YOUR website!
if (a+2)x^2- 2ax - a = 0
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what is range of a ?
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If you were to solve this problem by the quadratic formula,
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A = (a+2), B = -2a, C = -a
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the expression underneath the square root, B2 - 4AC, known as the
'discriminant', must not be negative.
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` ` ` ` ` `B2 - 4AC >= 0
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(-2a)2 - 4(a+2)(-a) >= 0
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` ` ` 4a2 + 4a(a+2) >= 0
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` ` `4a2 + 4a2 + 8a >= 0
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` ` ` ` ` `8a2 + 8a >= 0
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` ` ` ` ` 8a(a + 1) >= 0
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The zeros of the left side are 0 and -1
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Make a number line, and mark these solid since they are solutions.
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----------•--•------------
-4 -3 -2 -1 `0 +1 +2 +3 +4
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Choose any value in the leftmost region, the region left of -1. The easiest
value to choose is -2. Substitute it into the expression
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8a(a + 1)
8(-2)(-2 + 1) = -16(-1) = +16. This is nonnegative, so shade all those values.
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<==========•--•------------
`-4 -3 -2 -1 0 +1 +2 +3 +4
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Choose any value in the middle region, the region between -1 and 0. The
easiest value to choose is -1/2. Substitute it into the expression
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8a(a + 1)
8(-1/2)(-1/2 + 1) = -4(1/2) = -2. This is negative, so do not shade the values
in the middle region.
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Choose any value in the rightmost region, the region right of 0. The easiest
value to choose is 1. Substitute it into the expression
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8a(a + 1)
8(1)(1 + 1) = 8(2) = +16. This is nonnegative, so shade all those values.
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<==========•--•============>
`-4 -3 -2 -1 `0 +1 +2 +3 +4
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The interval notation for this is
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(-¥,-1] U [0, ¥)
`
Edwin
AnlytcPhil@aol.com
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