Question

Determine the best method to solve the following equation, then solve. 3t^2-5t+7=-3

Answer 1

The best method to solve the equation is employing the Quadratic formula method

which is,

`x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}`

The equation given is,

`3t^2-5t+7=-3`

Add both sides by 3

`\begin{gathered} 3t^2-5t+7+3=-3+3 \\ 3t^2-5t+10=0 \end{gathered}`

Then, solve with the quadratic formula

`t_(1,\: 2)=(-\left(-5\right)\pm√(\left(-5\right)^2-4\cdot\:3\cdot\:10))/(2\cdot\:3)`

Simplify the formula above

`t_(1,2)=\frac{5\pm\sqrt[]{25-120}}{6}=\frac{5\pm\sqrt[]{-95}}{6}`

Note that

`\sqrt[]{-95}=\sqrt[]{-1}*\sqrt[]{95}=\sqrt[]{95}i`

Therefore,

`t_(1,\: 2)=(5\pm√(95)i)/(6)`

Separate the solution

`t_1=(5+√(95)i)/(6),\: t_2=(5-√(95)i)/(6)`

Rewrite the solution in standard complex form

`t_1=(5)/(6)+(√(95))/(6)i,t_2=(5)/(6)-(√(95))/(6)i`

Hence, the solutions to the quadratic equation are

`t_1=(5)/(6)+i\frac{\sqrt[]{95}}{6},t_2=(5)/(6)-i\frac{\sqrt[]{95}}{6}`