# SOLUTION: if (a+2)x^2- 2ax - a = 0 what is range of a ?

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 Click here to see ALL problems on Polynomials-and-rational-expressions Question 16536: if (a+2)x^2- 2ax - a = 0 what is range of a ?Answer by AnlytcPhil(1276)   (Show Source): You can put this solution on YOUR website! if (a+2)x^2- 2ax - a = 0 ` what is range of a ? ` If you were to solve this problem by the quadratic formula, ` A = (a+2), B = -2a, C = -a ` the expression underneath the square root, B2 - 4AC, known as the 'discriminant', must not be negative. ` ` ` ` ` ` `B2 - 4AC >= 0 ` (-2a)2 - 4(a+2)(-a) >= 0 ` ` ` ` 4a2 + 4a(a+2) >= 0 ` ` ` `4a2 + 4a2 + 8a >= 0 ` ` ` ` ` ` `8a2 + 8a >= 0 ` ` ` ` ` ` 8a(a + 1) >= 0 ` The zeros of the left side are 0 and -1 ` Make a number line, and mark these solid since they are solutions. ` ----------•--•------------ -4 -3 -2 -1 `0 +1 +2 +3 +4 ` Choose any value in the leftmost region, the region left of -1. The easiest value to choose is -2. Substitute it into the expression ` 8a(a + 1) 8(-2)(-2 + 1) = -16(-1) = +16. This is nonnegative, so shade all those values. ` <==========•--•------------ `-4 -3 -2 -1 0 +1 +2 +3 +4 ` 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 ` 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. ` Choose any value in the rightmost region, the region right of 0. The easiest value to choose is 1. Substitute it into the expression ` 8a(a + 1) 8(1)(1 + 1) = 8(2) = +16. This is nonnegative, so shade all those values. ` <==========•--•============> `-4 -3 -2 -1 `0 +1 +2 +3 +4 ` The interval notation for this is ` (-¥,-1] U [0, ¥) ` Edwin AnlytcPhil@aol.com