Question 1151314
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If r and {{{cross(a)}}} s represent the solutions of the equation (3k + 14) * k = 5 and r > s, what is the value of the difference of r - s?
{{{(3k + 14) * k = 5}}}<br>
{{{3k^2+14k = 5}}}<br>
{{{3k^2+14k-5=0}}}<br>
{{{(3k-1)(k+5) = 0}}}<br>
{{{k = 1/3}}} or {{{k = -5}}}<br>
So r = 1/3 and s = -5; and the difference r-s is 1/3+5 = 16/3.<br>
Alternatively, we can find the value of r-s without factoring.<br>
The solutions to the equation<br>
{{{ax^2+bx+c=0}}}<br>
are<br>
{{{(-b+sqrt(b^2-4ac))/(2a)}}} and {{{(-b-sqrt(b^2-4ac))/(2a)}}}<br>
The difference between the two roots is then<br>
{{{2(sqrt(b^2-4ac))/(2a) = (sqrt(b^2-4ac))/a}}}<br>
For the quadratic in this problem that difference is<br>
{{{sqrt(14^2-4*3*(-5))/3= sqrt(256)/3 = 16/3}}}<br>