Question

find the value of A . I want answer with step by step explaination.​

Answer 1

Question

$\sf{If \: \: \dfrac{a³+3a}{3a²+1}=\dfrac{1241}{217} \: \: \: \: then \ the \ value\ of\ a}$

a)16

b)18

c)17

d)15

$\tt{Answer=\green{17}}$

Step by step explanation

$\sf{ \Longrightarrow \dfrac{a³+3a}{3a²+1}=\dfrac{1241}{217} }$

$\sf{ \Longrightarrow \dfrac{a(a^{2} +3)}{3a²+1}=\dfrac{1241}{217} }$

$\sf{ \Longrightarrow 217a({a^{2} +3})={1241}(3 {a}^{2} + 1) }$

$\sf{\Longrightarrow (217)({a^{2} +3})={1241} \times 3 {a}^{2} + 1241}$

$\sf{ \Longrightarrow 217a \times {a^{2}} + 217a \times 3={1241}(3 {a}^{2} + 1)}$

$\sf{ \Longrightarrow217a^{3} + 651a=3723 {a}^{2} + 1241}$

$\sf{\Longrightarrow 217a^{3} + 651a - 3723 {a}^{2} - 1241= 0}$

• By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where p divides the constant term -1241 and q divides the leading coefficient 217. List all candidates $\frac{p}{q}.$

$\sf{\Longrightarrow ±\frac{1241}{217},±\frac{1241}{31},±\frac{1241}{7},±1241,±\frac{73}{217},±\frac{73}{31},±\frac{73}{7},±73,±\frac{17}{217},±\frac{17}{31},±\frac{17}{7},±17,±\frac{1}{217},±\frac{1}{31},±\frac{1}{7},±1}$

• Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

a=17

• $\sf{By\ Factor\ theorem \ , \alpha\ -k\ is\ a\ facto of\ the\ polynomial\ for\ each\ root\ k. }$

• $\sf{Divide\ 217\alpha ^{3}-3723\alpha ^{2}+651\alpha -1241 by \alpha -17 to get 217\alpha ^{2}-34\alpha +73. }$

• $\sf{Solve\ the\ equation\ where\ the\ result\ equals\ to\ 0.}$

• $\sf{217\alpha ^{2}-34\alpha +73=0 }$

All equations of the form ax²+bx+c=0 can be solved using the quadratic formula:

$\sf{ \frac{-b±\sqrt{b^{2}-4ac}}{2a}.}$

$\sf{Substitute\ 217\ for\ a,\ -34 for\ b\ and\ 73 for\ c\ in\ the\ quadratic\ formula.}$

$\sf{\alpha =\frac{-\left(-34\right)±\sqrt{\left(-34\right)^{2}-4\times 217\times 73}}{2\times 217}}$

$\sf{=\alpha =\frac{34±\sqrt{-62208}}{434}}$

• As the square root of a negative number is not defined in the real field, there are no solutions.

$\sf{\alpha \in \emptyset }$

• List all found solutions.

a =17

$\\ \\ \sf{verification_{\ \ \ \ \Longrightarrow \dfrac{a³+3a}{3a²+1}=\dfrac{1241}{217}} }$

$\sf{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \Longrightarrow \dfrac{17³×3×17}{3×17²+1}=\dfrac{1241}{217} }$

$\sf{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \Longrightarrow \dfrac{4913×51}{3×289+1}=\dfrac{1241}{217} }$

$\sf{\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \Longrightarrow \dfrac{4964}{867+1}=\dfrac{1241}{217} }$

$\sf{\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \Longrightarrow \dfrac{4964}{868}=\dfrac{1241}{217} }$

$\sf{\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \Longrightarrow \dfrac{\cancel{4964}¹²⁴¹}{\cancel{868}²¹⁷}=\dfrac{1241}{217} }$

$\sf{\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \Longrightarrow \dfrac{1241}{217}=\dfrac{1241}{217} }$

$\sf{\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \Longrightarrow LHS=RHS_{~~~\pink{VERIFIED}}}$

Answer 2

Question

$\sf \large \: \dfrac{ {a}^{3} + 3a }{3 {a}^{2} + 1} = \dfrac{1241}{217}$

$\sf\huge\bold{\underline{\underline{{Solution}}}}$

$\sf \large \: \dfrac{ {a}^{3} + 3a }{3 {a}^{2} + 1} = \dfrac{1241}{217}$

Now ,cross multiply:

$\sf \longmapsto217 {a}^{3} + 651a = 3723 {a}^{2} + 1241$

$\sf \longmapsto217 {a}^{3} + 651a - 3723 {a}^{2} - 1241 = 0$

$\sf \longmapsto217 {a}^{3} - 3723 {a}^{2} + 651a - 1241 = 0$

$\sf \longmapsto217 {a}^{3} - 3689 {a}^{2} - 34 {a}^{2} + 578a + 73a - 1241 = 0$

$\sf \longmapsto217 {a}^{2} (a - 17) - 34a(a - 17) + 73(a - 17) = 0 = 0$

$\sf \longmapsto(a - 17)\bold{(217 {a}^{2} - 34a + 73) }= 0$

$\sf a - 17 = 0$

$\sf \bold{ \red{a = 17}}$

217a²-34a+73

$\sf a = \dfrac{ - b \pm \sqrt{ {b}^{2} - 4ac} }{2a}$

$\sf a = \dfrac{34 \pm \sqrt{ {( - 34)}^{2} - 4(217)(73)} }{2 \times 217}$

$\sf a = \dfrac{34 \pm \sqrt{1156 - 63364} }{434}$

$\sf a = \dfrac{34 \pm \sqrt{ - 62208} }{434}$

$\sf a = \dfrac{34 \pm \sqrt{62208i} }{434}$

Second part is in the form of a quadratic equation so it can be solve through quadratic formula:

We obtained :

$\sf \large \: a = \dfrac{34 \pm \sqrt{62208i} }{434}$

The second value of a which we obtained from quadratic formula is imaginary means not a real value.So,the real value of a is 17

$\sf\large\bold{∴}$ $\sf\large\bold{\red{a=17}}$