Question

How do you solve `30 + x − x^2 = 0`?

Answer 1

`x=-5,6`

Explanation:

Invert (multiply by -1, has the same solutions) and complete the square:
`x^2-x-30=(x-1/2)^2-121/4=0`

Solve for `x`:
`(x-1/2)^2=121/4`

=>

`x-1/2=+-11/2`

=>

`x=(1+-11)/2`

Answer 2

solve `y = -x^2 + x + 30 = 0`

Ans: -5 and 6

Explanation:

I use the new Transforming Method (Google, Yahoo, Bing Search)
Find 2 numbers knowing sum (1) and product (-30). Roots have opposite signs since a and c have opposite signs.
Factor pairs of (-30) --> (-2, 15)(-4, 5)(-5, 6). This sum is 1 = b.
Since a < 0. then the 2 real roots are: -5 and 6.

Answer 3

You could use the quadratic formula.

Explanation:

First, rewrite your quadratic in the form

`color(blue)(ax^2 + bx + c = 0)`

for which the quadratic formula takes the form

`color(blue)(x_(1,2) = (-b +- sqrt(b^2 - 4ac))/(2a)`

You will start from

`-x^2 + x + 30= 0`

which can be rewritten as

`-(x^2 - x - 30) = 0`

In this case, `a=11`, `b=-1`, and `c=-30`.

The two solutions to this quadratic equation will thus be

`x_(1,2) = (-(-1) +- sqrt( (-1)^2 - 4 * (1) * (-30)))/(2 * (1))`

`x_(1,2) = (1 +- sqrt(121))/(-2) = (1 +-11)/2`

`x_1 = (1 + 11)/(2) = color(green)(6)`

`x_2 = (1 - 11)/(2) = color(green)(-5)`