HOW TO FIND WHETHER THE PARABOLA INTERSECTS X AXIS WITHOUT GRAPH

We may find number of points where the parabola intersects the x-axis, using the formula for discriminant.

Question 1 :

Without sketching the graphs, find whether the graphs of the following functions will intersect the x-axis and if so in how many points.

(i)   y = x² + x + 2

Solution :

Discriminant  =  b² − 4ac

a =  1, b = 1 and c = 2

b² − 4ac  =  (1)² − 4(1)(2)

  =  1 - 8

  =  -7 < 0

Does not intersect the x-axis.

(ii) y = x² − 3x − 7

Solution :

Discriminant  =  b² − 4ac

a =  1, b = -3 and c = -7

b² − 4ac  =  (-3)² − 4(1)(-7)

  =  9 + 28

  =  36 > 0

Hence the curve intersects x-axis at two points.

(iii) y = x² + 6x + 9

Solution :

Discriminant  =  b² − 4ac

a =  1, b = 6 and c = 9

b² − 4ac  =  (6)² − 4(1)(9)

  =  36 - 36

  =  0 

Hence the curve does not intersect x-axis at any point.

Question 2 :

Write f(x) = x² + 5x + 4 in completed square form.

Solution :

  =   x² + 5x + 4

Multiply and divide the coefficient of x by 2.

  =   x² + (2/2) ⋅ 5 ⋅ x + 4

  =   x² + 2 ⋅ x ⋅ (5/2) + (5/2)² - (5/2)² + 4

  =   (x + (5/2))² - (25/4) + 4

  =   (x + (5/2))² + (16 - 25)/4

  =   (x + (5/2))² + (-9)/4

  =   (x + (5/2))² - (3/2)²

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