How many intercepts does #y = x^2 − 5x + 6# have?

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Answer 1 · 2015-08-13T00:00:37

Two #x#-intercepts and one #y#-intercept.

You can find this function's #x#-intercepts by making equal to zero and its #y#-intercepts by evaluating the function for #x=0#.

For the #x#-intercepts, you have

#x^2 - 5x + 6 = 0#

You can determine how many solutions this quadratic has by calculating its discriminant, #Delta#, which, for a general form quadratic equation

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

takes the form

#color(blue)(Delta = b^2 - 4ac)#

In your case, the discriminant will be

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

#Delta = 25 - 24 = color(green)(1)#

When #Delta>0#, the quadratic equation has two distinct real roots that take the form

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

In your case, these roots will be

#x_(1,2) = (-(-5) +- sqrt(1))/(2 * 1)#

#x_(1,2) = (5 +- 1)/2 = {(x_1 = (5 + 1)/2 = 3), (x_2 = (5-1)/2 = 2) :}#

This means that the function will have two #x#-intercepts at #x=2# and #x=3#.

The #y#-intercept will be

#y = (0)^2 - 5 * (0) + 6 = 6#

The function will intercept the #x#-axis in the points #(2,0)# and #(3,0)#, and the #y#-axis in the point #(0, 6)#.

graph{x^2 - 5x + 6 [-10, 10, -5, 5]}