Express Yourself

Algebra Level 2

If $a$ is a complex number such that $a^2 = 2a - 3$ , which of the following expressions equals $4$ ?

\dfrac{a^2+3}{2a}

\left ( \dfrac{a^2}{a-\tfrac{3}{2}} \right )^2

(a^2-2a-2)(a^2-2a+2)

a^3

+

1

1 solution

U Z
Jan 3, 2015

$a^2 = 2(a - \dfrac{3}{2})$

$\dfrac{a^2}{a - \dfrac{3}{2}} = 2$

$\Bigg(\dfrac{a^2}{a - \frac{3}{2}} \Bigg)^2 = 4$

How about the other three expressions?

Guilherme Dela Corte - 6 years, 5 months ago

$a^2 - 2a = -3$

$\implies (a^2 - 2a)^2 = 9$

$(a^2-2a-2)(a^2-2a+2) = (a^2 - 2a)^2 - 4 = 5$

$a^2 + 3 = 2a$

$\implies \dfrac{a^2 - 3}{2a} =1$

$a = re^{ix}$

$a^3 + 1 = r^3e^{3ix} + 1~which~will~be~a~complex~number$

U Z - 6 years, 5 months ago