Answer :
Answer:
d . 232
Step-by-step explanation:
The first equation is
[tex]3 {x}^{2} - 2x + c = 0[/tex]
Where a=3, and b=-2.
Let m and n be the roots of this equation.
Then the difference between the roots is
[tex]m - n = \sqrt{ {(m + n)}^{2} - 4mn} [/tex]
[tex]m - n = \sqrt{ {( \frac{ - 2}{3}) ^{2} - 4( \frac{c}{3} )} } [/tex]
[tex]m - n = \sqrt{ {\frac{ 4}{9}- 4( \frac{c}{3} )} } [/tex]
The second equation is
[tex]2 {x}^{2} - cx + 3 = 0[/tex]
a=2, b=-c and c=3
Let m and n be roots, then;
[tex]m - n = \sqrt{ {(m + n)}^{2} - 4mn} [/tex]
[tex]m - n = \sqrt{ {( - \frac{c}{2} )}^{2} - 4( \frac{3}{2}) } [/tex]
[tex]m - n = \sqrt{ \frac{ {c}^{2} }{4} - 6} [/tex]
We equate the two expressions to get:
[tex] \sqrt{ \frac{ {c}^{2} }{4} - 6} = \sqrt{ \frac{4}{9} - \frac{4c}{3} } [/tex]
[tex]\frac{ {c}^{2} }{4} - 6 = \frac{4}{9} - \frac{4c}{3} [/tex]
Multiply through by 36;
[tex]36 \times \frac{ {c}^{2} }{4} - 6 \times 36 = \frac{4}{9} \times 36 - \frac{4c}{3} \times 36[/tex]
[tex]9 {c}^{2} - 216 =16 - 48c[/tex]
[tex]9 {c}^{2} + 48 =16 + 216 [/tex]
[tex]9 {c}^{2} + 48 =232[/tex]