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}}}.
<pre>{{{matrix(1,3, 2ax^2 - 4ax + a + 1, "=", 0)}}} =====> {{{matrix(1,4, (2a)x^2 (- 4a)x, " +(a + 1)", "=", 0)}}}

Let smaller root be R
Then larger root = 5R
Sum of the roots, or {{{matrix(1,9, s, "=", - b/a, "=", - (- 4a)/(2a), "=", 4a/(2a), "=", 2)}}}. Also, sum of roots = R + 5R = 6R.
                                  Therefore, {{{system(matrix(2,3, 2, "=", 6R, 2/6, "=", R), matrix(1,5, 1/3, "=", R, "(smaller", "root)"))}}}

Product of the roots, or {{{matrix(1,5, p, "=", c/a, "=", (a + 1)/(2a))}}}. Also, product of roots = R(5R) = 5R<sup>2</sup> = {{{matrix(1,5, 5(1/3)^2, "=", 5(1/9), "=", 5/9)}}} 
                       Therefore, {{{matrix(1,3, (a + 1)/(2a), "=", 5/9)}}}
                                     10a = 9(a + 1)----- Cross-multiplying
                                     10a = 9a + 9
                                10a - 9a = 9
                                      {{{highlight_green(matrix(1,3, a, "=", 9))}}}
***BTW, this is Part (a).</pre>