Alan

This can be factored as (2x - 1)(x³ - 3) so there is a zero at the positive rational value 1/2 and at the positive irrational value ³√3.   The other two roots are complex.  There are no negative real roots.

.

Use a ratio approach:

X/100 = 500/30

X = 100*500/30 

$$X=1666\frac{2}{3}$$

No.  The y-component of velocity is zero at the highest point, but the x-component is still non-zero, hence the resultant velocity is still non-zero.

.

.Melody, Geogebra is certainly able to draw ellipses:

Do you mean something like:

(Probably not, as rounding to three decimal places seems irrelevant!).

.

Volume is (4/3)*pi*radius³ and a litre is 10^-3m³

So Volume = (4/3)*pi*7.7³ m³ ≈ 1912 m³ = 1.912*10⁶L

Hmm.  Plenty of answers.  Luckily they're all the same!

.

This might be asking for the escape velocity of the Moon relative to that of Earth.  If so, escape velocity is given by

$$v_e=\sqrt{\frac{2GM}{r}}$$

where G = Newtonian gravitaional constant

M is mass

r is radius

So    $$\frac{v_{eMoon}}{v_{eEarth}}=\sqrt{\frac{M_{Moon}r_{Earth}}{M_{Earth}r_{Moon}}}$$

or     $$\frac{v_{eMoon}}{v_{eEarth}}=\sqrt{\frac{3.7}{81}}=0.2137$$

.

In general, the roots of a function, f(x), say, are the values of x that make f(x) zero.

.

See if the following graph helps

.

Nice illustration Melody.

.