Question 1090463
<br>When the ball hits the ground, its height is 0.  So you want to solve the equation
{{{-16t^2-18t+405=0}}}
The question asks for an answer to the nearest tenth, suggesting that the answer is not a "nice" number and has to be rounded.  However, the quadratic expression factors, yielding an exact answer.<br>
You could just use the quadratic formula, or perhaps a graphing calculator or some other such tool, to find the answer.  But let's instead get some practice with this problem on factoring quadratics.<br>
I would first factor out a -1, so that the leading coefficient is positive; factoring a quadratic with a negative leading coefficient I find to be a very unpleasant task.  So our quadratic is
{{{-1(16t^2+18t-405)}}}<br>
With a leading coefficient of 16, the leading coefficients of our two binomial factors could be 1 and 16, or 2 and 8, or 4 and 4.<br>
But in fact they can't be 4 and 4 -- because then the coefficient of the linear (x) term would have to be a multiple of 4, which it is not.<br>
And leading coefficients of 1 and 16 on the binomial factors is unlikely, so I would look first for a factorization with 2 and 8 as the leading coefficients of the two binomial factors.<br>
Some of the possible factorizations of the constant term 405 are 81*5, 27*15, 9*45, ....  Some playing around with the different possible combinations shows the factorization to be
{{{16t^2+18r-405=(8t+45)(2t-9)}}}<br>
Setting the first factor equal to 0 gives us a negative value for the time t, which does not make sense in the actual problem.  Setting the second factor equal to 0 gives us the value of 4.5 for t.<br>
So that is the answer to the problem: the ball hits the ground after 4.5 seconds.