Question

Consider the two equations below. Explain completely the similarities and

differences in how you would solve each equation. Be clear and complete.

3* = 12 and x3 = 12

Answer 1

Here, the given equations,

`3^x=12` and
`x^3=12`

Similarity: In both equations there is only one variable ( i.e. x)

Difference:
`3^x=12` is an exponential equation while
`x^3=12` is a polynomial equation.

Now, when we solve an exponential equation we take log in both sides of the equation as follows:

`3^x=12`

`\log(3^x)=\log12`

`x\log 3 =\log 12` ( ∵
`\log m^n=n\log m` )

`\implies x =(\log 12)/(\log 3)`

Hence, the solution of the equation
`3^x=12` is
`x=(\log 12)/(\log 3)`.

While, when we solve a polynomial we find the roots as follows:

`x^3=12`

`x^3-12=0`

`x^3-(12^(1)/(3))^3=0`

`(x-12^(1)/(3))(x^2+12^(1)/(3)x+(12^(1)/(3))^2)=0`

By zero product property,

`(x-12^(1)/(3))=0` or
`(x^2+12^(1)/(3)x+(12^(1)/(3))^2)=0`

If
`(x-12^(1)/(3))=0`, then
`x=12^(1)/(3)`

If
`x^2+12^(1)/(3)x+(12^(1)/(3))^2=0`,

Then, by quadratic formula,

`x=\frac{-12^(1)/(3)\pm \sqrt{12^(2)/(3)-4(1)(12^(1)/(3))^2}}{2}`

`=\frac{-12^(1)/(3)\pm \sqrt{12^(2)/(3)-4(12^(2)/(3))}}{2}`

`=\frac{-12^(1)/(3)\pm i\sqrt{3(12^(2)/(3))}}{2}`

`=12^(1)/(3)((-1\pm i√(3))/(2))`

Hence, the solutions of the equation
`x^3=12` are
`12^(1)/(3)`,
`12^(1)/(3)((-1+i√(3))/(2))` and
`12^(1)/(3)((-1-i√(3))/(2))` .