Question

How do you solve the quadratic `7x^2+5=-137` using any method?

Answer 1

There are no real solutions. But you can solve this using complex numbers. Then, the solutions (both imaginary) are:

`x_1 = + sqrt {142/7} cdot i and x_2 = - sqrt {142/7} cdot i`

Explanation:

You can probe there are no solutions in `RR` clearing the `x` as follows:

`7 x^2 + 5 = - 137 color(white) "." rArr color(white) "." 7 x^2 = - 137 - 5 = - 142`,

then

`7 x^2 = - 142 color(white) "." rArr color(white) "." x^2 = - 142/7 color(white) "." rArr color(white) "." x = +- sqrt {- 142/7} !in RR`.

However, if we still want to give an answer, we can use complex numbers using the definition of the imaginary number `i`:

Since `i = sqrt {- 1}` we can write the solutions of the quadratic equation of the form:

`x = +- sqrt {- 142/7} = +- sqrt {142/7} cdot sqrt {- 1} = +- sqrt {142/7} cdot i in CC`.