Final answer:
The solution of the linear-quadratic system
and y - x = 2 has two real solutions: (-5, -3) and (1, 3). The quadratic equation
has one repeated real solution: x = 4. To simplify the expression -5 + i / 2i, multiply by the conjugate to get -5 + i/2.
Step-by-step explanation:
The solution of the linear-quadratic system of equations
and y - x = 2 is determined by substitution or elimination. For the first equation by substituting y from the second into the first, you get
, which simplifies to
. This can be factored to (x + 5)(x - 1) = 0, giving solutions x = -5 or x = 1.
Plugging these back into y - x = 2 gives the corresponding y values as y = -3 and y = 3, respectively. Therefore, the system has two real solutions: (-5, -3) and (1, 3), which corresponds to option C.
For the second part, the quadratic equation
can be simplified by dividing each term by 2, yielding
, which is a perfect square trinomial.
Factoring gives (x - 4)(x - 4) = 0, so the only solution is x = 4. However, since the question suggests there are two values, there might be a misunderstanding or typo. In canonical form, if we perceive the quadratic as
, the answer is still x = 4, which is repeated, and option A is the closest to this result.
To simplify the complex expression, -5 + i / 2i, multiply the numerator and denominator by the conjugate of the denominator, in this case, -2i.
It becomes (-5 + i)(-2i) / (2i)(-2i) = (10i - 2) / -4 = -5i/2 + 1/2, which corresponds to option B, -5 + i/2.