SOLUTION: Find all solutions (including solutions that are not real) to the following equations: x2 – 144 = 0 x4 – 81 = 0 x3 – 27 = 0 z3 + 8 = 0

Algebra ->  Systems-of-equations -> SOLUTION: Find all solutions (including solutions that are not real) to the following equations: x2 – 144 = 0 x4 – 81 = 0 x3 – 27 = 0 z3 + 8 = 0       Log On


   



Question 923715: Find all solutions (including solutions that are not real) to the following equations:
x2 – 144 = 0
x4 – 81 = 0
x3 – 27 = 0
z3 + 8 = 0

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2+-144+=+0
x%5E2+-12%5E2+=+0
%28x-12%29%28x%2B12%29+=+0
solutions:
if %28x-12%29=+0=> highlight%28x=12%29
if %28x%2B12%29=+0=> highlight%28x=-12%29


x%5E4+-81+=+0
x%5E4+-3%5E4+=+0
%28x%5E2%29%5E2+-%283%5E2%29%5E2+=+0
%28x%5E2%29-%283%5E2%29%28x%5E2%29%2B%283%5E2%29+=+0
%28x-3%29%28x+%2B3%29%28x-3i%29%28x%2B3i%29+=+0
solutions:
real:
if %28x-3%29+=+0=> highlight%28x=3%29
if %28x%2B3%29=+0=>highlight%28x=-3%29
complex:
if %28x-3i%29=+0=>highlight%28x=3i%29
if %28x%2B3i%29=+0=>highlight%28x=-3i%29


x%5E3+-27+=+0
x%5E3+-3%5E3+=+0
%28x-3%29%28x%5E2%2B3x%2B9%29+=+0
one solution: %28x-3%29=0=>highlight%28x=3%29
for +%28x%5E2%2B3x%2B9%29+=+0 use quadratic formula:
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
x+=+%28-3+%2B-+sqrt%28+3%5E2-4%2A1%2A9+%29%29%2F%282%2A1%29+
x+=+%28-3+%2B-+sqrt%28+9-36+%29%29%2F2+
x+=+%28-3+%2B-+sqrt%28+-25+%29%29%2F2+
x+=+%28-3+%2B-+5i%29%2F2+
complex solutions:
x+=+%28-3+%2B+5i%29%2F2+
highlight%28x+=+-3%2F2+%2B+5i%2F2+%29
and
x+=+%28-3+-+5i%29%2F2+
highlight%28x+=+-3%2F2+-+5i%2F2%29+




z%5E3+%2B+8+=+0
z%5E3+%2B+2%5E3+=+0
%28z%2B2%29+%28z%5E2-2z%2B4%29+=+0

one solution: %28x%2B2%29=0=>highlight%28x=-2%29
for +%28z%5E2-2z%2B4%29=+0 use quadratic formula:
z+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
z=+%28-%28-2%29+%2B-+sqrt%28+%28-2%29%5E2-4%2A1%2A4+%29%29%2F%282%2A1%29+
z+=+%282+%2B-+sqrt%284-16+%29%29%2F2+
z+=+%282+%2B-+sqrt%28+-12+%29%29%2F2+
z+=+%282+%2B-+sqrt%28+-4%2A3+%29%29%2F2+
z+=+%282+%2B-+2i%2Asqrt%283%29%29%2F2+
complex solutions:
x+=+%282+%2B+2i%2Asqrt%283%29%29%2F2+
x+=+2%2F2+%2B+2i%2Asqrt%283%29%2F2+
highlight%28x+=+1%2B+sqrt%283%29%2Ai%29+
and
x+=+%282+-2i%2Asqrt%283%29%29%2F2+
x+=+2%2F2+-+2i%2Asqrt%283%29%2F2+
highlight%28x+=+1-+sqrt%283%29%2Ai%29+