Question

How do you solve the quadratic `7n^2+6=-40` using any method?

Answer 1

`x = (+- isqrt322)/7`

Explanation:

`7n^2 + 6 = -40`

To use the quadratic formula to find the zeroes, we need to make sure the equation is written in the form `color(red)(a)x^2 + color(magenta)(b)x + color(blue)(c) = 0`.

To do so, we need to add `color(orange)40` to both sides of the equation:
`7n^2 + 6 quadcolor(orange)(+quad40) = -40 quadcolor(orange)(+quad40)`

`7n^2 + 46 = 0`

So we know that:
`color(red)(a = 7)`

`color(magenta)(b = 0)`

`color(blue)(c = 46)`

The quadratic formula is `x = (-color(magenta)(b) +- sqrt(color(magenta)(b)^2 - 4color(red)(a)color(blue)(c)))/(2color(red)(a))`.

Now we can plug in the values for `color(red)(a)`, `color(magenta)(b)`, and `color(blue)(c)` into the quadratic formula:

`x = (-color(magenta)(0) +- sqrt((color(magenta)(0))^2 - 4(color(red)(7))(color(blue)(46))))/(2(color(red)(7)))`

Simplify:
`x = (+- sqrt(-1288))/14`

The solution is imaginary since we cannot take the square root of a negative number.

However, we know that `i`, or imaginary, is equal to `sqrt(-1)`. Therefore, we can factor out a `-1` from the square root and make it an `i`:

`x = (+-isqrt(1288))/14`

`x = (+- 2isqrt(322))/14`

`x = (+- isqrt322)/7`

This is the same thing as:
`x = (isqrt322)/7` and `x = (-isqrt322)/7`
because `+-` means "plus or minus."

To clarify, this doesn't mean that there is a zero or root. This is an imaginary solution, meaning that there are no zeros. To prove this, let's look at the graph of this equation:

(desmos.com)

As you can see, there are no zeroes. The vertex starts above the `x`-axis.

Hope this helps!