Question 1195204
.
Solve these two equations:


(a)   The quadratic equation {{{2ax^2-4ax+a+1=0}}} has two roots. 
        If one root is five times the other, what is the value of a?


(b)   If {{{x+2}}} and {{{x-3}}} are factors of the polynomial {{{p(x) = x^3+5x^2+ax+b}}}, 
        find {{{a}}}.
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<H3>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Part (a)</H3>

<pre>
Let z be one root of the equation; then 5z is another root.


Apply the Vieta's theorem.  it gives you these two equations for unknowns "a" and "z":


    (1)  the sum of the roots z and 5z is equal to {{{-(-4a)/(2a)}}} = 2

             z + 5z = 2,  or  6z = 2,  z = {{{2/6}}} = {{{1/3}}}.


    (2)  the product of the roots is equal to {{{(a+1)/(2a)}}}

             z*(5z) = {{{(a+1)/(2a)}}},  which implies  {{{(1/3)*(5/3)}}} = {{{(a+1)/(2a)}}},

             {{{5/9}}} = {{{(a+1)/(2a)}}},  5*(2a) = 9*(a+1),  10a = 9a + 9,  a = 9.


Thus we get the <U>ANSWER</U> :  a = 9.
</pre>

Part &nbsp;(a) &nbsp;is solved.



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