SELECT THE CORRECTOR ALTERNATIVES(ONE OR MORE THAN ONE CORRECT ANSWERS If a is a root of the equation $2x(x+1)=1,$ then the other root is.
- $3{a^2} - 4a$
- $ - 2a\left( {a + 1} \right)$
- $4{a^3} - 3a$
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None of these
The equation is 2x^2 + 2x - 1 = 0. If a is a root, then 2a^2 + 2a - 1 = 0. The sum of roots is -b/a = -2/2 = -1, so the other root is -1 - a. None of the provided options A, B, or C simplify to -1 - a.
Rearranging the equation 2x times the sum of x and 1 equals 1 gives the quadratic 2x squared plus 2x minus 1 equals 0. Since a is a root of this equation, it satisfies 2a squared plus 2a minus 1 equals 0, which rearranges to a squared equals 1 divided by 2 minus a. If a is one root, the other root of the quadratic can be found by Vieta's formulas; the sum of the roots is negative 1, so the other root is negative 1 minus a. Evaluating the first option, 3a squared minus 4a equals 3 times 1 divided by 2 minus a minus 4a, which simplifies to 3 divided by 2 minus 7a. Since a is irrational, this is not equal to negative 1 minus a. Evaluating the second option, negative 2a times the sum of a and 1 equals negative 1, which is also not equal to the second root. Evaluating the third option, 4a cubed minus 3a equals a times 4a squared minus 3. Substituting a squared gives a times 2 minus 4a minus 3, which is negative a times 4a plus 1. Since 2a squared is 1 minus 2a, this simplifies to a squared, which is not the second root. Therefore, none of the provided options represent the other root.