Question

How do you solve `7x^2=-21`?

Answer 1

See a solution process below:

Explanation:

First, divide each side of the equation by `color(red)(7)` to isolate the `x^2` term while keeping the equation balanced:

`(7x^2)/color(red)(7) = -21/color(red)(7)`

`(color(red)(cancel(color(black)(7)))x^2)/cancel(color(red)(7)) = -3`

`x^2 = -3`

Next, we would take the square root of each side of the equation to solve for `x` while keeping the equation balanced.

However, there is no Real solution for the square root of a negative number, in this case the `sqrt(-3)`

Therefore, there is no real solution or the solution set is the empty or null set: `{O/}`

Answer 2

No real solutions but rather, two complex solution : `x=isqrt3,x=-isqrt3`

Explanation:

Divide `7` to both sides;

`cancel(7/7)x^2=-21/7`

`x^2=-3`

`sqrt(x^2)=sqrt(-3` (Apply the square root property)

`x=+-sqrt(-3)`

There are no "real solutions" since the number inside the radical is negative but there are two "complex" solutions:

We can rewrite the expression above as:

`x=sqrt(-1)*sqrt3, x=-sqrt(-1)*sqrt3`

*Recall that `sqrt-1=i`

Therefore we can simplify the expression as:

`x=isqrt3,x=-isqrt3` (This is our final answer)