Question

How can you tell from the standard form whether a quadratic function has exactly one zero

Answer 1

The discriminant equals zero.

Explanation:

The standard form of a quadratic equation is

ax² + bx + c = 0

The solution is the familiar quadratic formula

`x = (-b\pm√(b^2-4ac))/(2a)`

The term b² -4ac is the discriminant (D).

D tells you the number of roots.

If D = 0, the solution to the equation becomes

`x = (-b\pm√(0))/(2a)`

`x = (-b)/(2a)`

If D = 0, there is exactly one zero.

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Example:

Find the zeroes of

f(x) = x² - 2x + 1

`x = \frac{2 \pm \sqrt{(-2)^(2) - 4(1)(1)}}{2(1)}`

`x = (2 \pm√(4 - 4))/(2)`

`x = (2 \pm√(0))/(2)`

`x = (2 \pm 0)/(2)`

`x = (2)/(2)`

x = 1

The graph is a parabola with its vertex touching the x-axis at x = 1.