Without drawing the graph or completely solving the equation?

Question

Without drawing the graph or completely solving the equation, explain how to find the number of X intercepts of a quadratic function.

Answer 1

The number of intercepts with the x-axis a quadratic function has is also called the number of real roots. We can find how many real roots a quadratic function has, by calculating its discriminant.

For the example equation: ax^2 + bx + c = 0, the discriminant is b^2 - 4ac.

So, for example, take the equation 4x^2 + 2x + 5 = 0 (In comparison, a=4, b=2 and c=5)

The discriminant = 2^2 - (4*4*5) = 4 - 80 = -76

The rule is: if the discriminant is less than 0, there are no real roots (i.e. no x-axis intercepts)

if the discriminant is equal to 0, there is one real root, or a repeated real root.

if the discriminant is more than 0, there are two real roots (i.e. 2 x-axis intercepts).

So for the example function above, its discriminant was equal to -76, which is less than 0, so it has no real roots (no x-axis interceptions). It is said to have 2 imaginary roots instead.



Hope this helps!

Answer 2

There is no way to know, for most standard form quadratics,

unless you check the discriminant, b^2 - 4ac.



If the discriminant is < 0, there are no x-intercepts.



If the discriminant = 0, there is only one x-intercept (which is also the vertex).



If the discriminant is > 0, there are exactly two x-intercepts.





There are some tricks for some quadratics written in standard form.



If there is no constant term, only ax^2 + bx,

then the quadratic will have two x-intercepts (and one will always be zero).



If there is no middle term, only ax^2 + c, then the quadratic may have none or two.

If the sign of a and c are the same (both positive or both negative), there are none.

If the sign of a and c are different (one positive, one negative), then there are two.



If there is only the quadratic term, ax^2,

then there is only one x-intercept, (0, 0), which is also the vertex.

Answer 3

Look at the degree of the function. The highest degree tells you how many zeroes the function has.