Question

x3 – x2+x+46, if x = 2+3i.​

Answer 1

Answer:

$x^{3}-x^{2}+x+46=7$

If we substitute $x=2+3i$

Step-by-step explanation:

Given equation is:

$x^{3}-x^{2}+x+46=7\,\,\,\,\,eqn(1)$

Some formula of imaginary number is:

$i^2=-1\\i^3=-i$

Here we are substituting the values separately for making it easier.

$x^3=(2+3i)^3\\=(2)^3+(3i)^3+3\cdot 2^2\cdot 3i+3\cdot 2\cdot (3i)^2\\=8+27(-i)+36(i)+3\cdot 2\cdot 9(i)^2\\=8-27(i)+36(i)+54(-1)\\=8+9i-54\\=-46+9i$

Now solving for $x^2$.

$x^2=(2+3i)^2\\=4+9(i)^2+2\cdot 2\cdot 3i\\=4+9(-1)+12i\\=-5+12i$

Solving for x.

$x=2+3i$

Substituting all these values in eqn(1) we get:

$x^3-x^2+x+46=-46+(9i)-(-5+12i)+2+3i+46\\=(9i)+5-12i+2+3i\\=7$

So the given expression results us the answer 7.