How many times does 3x2-4x plus 5 cross the x-axis?

Answer 1

It does not cross the x axis, at all.

Let f(x) = 3x2 - 4x + 5.

The graph of f(x) crosses the x-axis if and only if 3x2 - 4x + 5 = 0, which implies:

x2 - (4/3)x + (5/3) = 0

x2 - (4/3)x + (4/9) + (5/3) = 4/9

[x - (2/3)]2 + (5/3) = 4/9

[x - (2/3)]2 = (4/9) - (5/3) = (4/9) - (15/9) = -11/9

x - (2/3) = sqrt(-11/9) or -sqrt(-11/9).

x = (2/3) plus or minus i*(1/3)*(sqrt(11)

Since these numbers are not real, the graph never crosses the x-axis.

Or here's some other other ways:

If you use the quadratic formula, look at the discriminant (b² - 4ac), which is the part under the square root. This equals (-4)² - 4*3*5 = -44. So you have the square root of a negative number, which is imaginary. The solution which makes the equation equal zero will have a real component, and an imaginary component, which is not 'on' the x-axis.

You could also use calculus and take the derivative, giving you y' = 6x -4, which will tell you that the vertex is at x = 2/3, Substituting this into the original, gives you y = 3 2/3. Since the x² term has a positive coefficient, the parabola opens up, therefore [ y = 3 2/3] is a minimum, which is above the x-axis, so it doesn't cross the x-axis.