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

2 Answers
Aug 2, 2015

Answer:

#Delta = 49#

Explanation:

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)#

Aug 2, 2015

Answer:

Solve: #x^2 - 5x = 6#

Explanation:

#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.