What is the discriminant of `x^2 - 5x = 6` and what does that mean?

For a quadratic equation that has the general form

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

the discriminant can be calculated by the formula

`color(blue)(Delta = b^2 - 4 * a * c)`

Rearrange your quadratic by adding `-6` to both sides of the equation

`x^2 - 5x - 6 = color(red)(cancel(color(black)(6))) - color(red)(cancel(color(black)(6)))`

`x^2 - 5x -6 = 0`

In your case, you have `a=1`, `b=-5`, and `c=-6`, so the discriminant will be equal to

`Delta = (-5)^2 - 4 * 1 * (-6)`

`Delta = 25 + 24 = 49`

SInce `Delta>0`, this quadratic equation will have two disctinct real solutions. Moreover, because `Delta` is a perfect square, those two solutions will be rational numbers.

The general form of the two solutions is given by the quadratic formula

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

In your case, these two solutions will be

`x_(1,2) = (-(-5) +- sqrt(49))/(2 * 1) = (5 +- 7)/2`

so that

`x_1 = (5 + 7)/2 = color(green)(6)` and `x_2 = (5-7)/2 = color(green)(-1)`

`y = x^2 - 5x - 6 = 0`
In this case, (a - b + c = 0), use the shortcut --> 2 real roots--> - 1 and `(-c/a = 6).`

REMINDER of SHORCUT

When (a + b + c = 0) --> 2 real roots: 1 and `c/a`
When (a - b + c = 0) --> 2 real roots: - 1 and `(- c/a)`
Remember this shortcut. It will save you a lot of time and effort.