For the equation $\mid x\mid ^{2} + \mid x \mid - 6 = 0$
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there is only one root
- the sum of the roots is $1$
- the sum of the roots is $0$
- the product of the roots is $4$
Reveal answer
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C
Correct answer
Explanation
Let u = |x|. The equation becomes u^2 + u - 6 = 0, which factors to (u+3)(u-2) = 0. Since u = |x| must be non-negative, u = 2. Thus |x| = 2, so x = 2 or x = -2. The sum of the roots is 2 + (-2) = 0.
AI explanation
Substitute y equals the absolute value of x into the equation to get the quadratic y squared plus y minus 6 equals 0. Factoring this gives y plus 3 times y minus 2 equals 0, yielding y equals 2 or y equals negative 3; since the absolute value of x cannot be negative, the absolute value of x equals 2. This gives two real roots for x, which are 2 and negative 2, making their sum equal to 0.