Two, one, or none

The idea

A parabola may cut the x-axis at two distinct points, at two coincident points, or not at all, so a quadratic polynomial has two distinct zeroes, two equal zeroes (one zero), or no zero.

How many?

A parabola is a cup. The x-axis is a straight line. How many times can the cup cross it?

A quadratic polynomial can have…

• always two zeroes
• 0, 1 or 2 zeroes
• exactly one zero

Lift and drop the cup

The curve is y = x² − 4x + c. Slide c to lift or drop the cup.

Three cases

A parabola may cut the x-axis at two distinct points, at two coincident points (it just touches), or not at all. So a quadratic polynomial has two distinct zeroes, two equal zeroes (one zero), or no zero.

So a polynomial of degree 2 has at most two zeroes.

Notes

A quadratic polynomial has two distinct zeroes, two equal zeroes (one zero) or no zero: its parabola cuts the x-axis at two points, touches it, or misses it.

Check yourself

A parabola lies completely above the x-axis. How many zeroes does the polynomial have?

A parabola just touches the x-axis at one point. What do the zeroes look like?

In y = x² − 4x + c, for which c does the curve touch the x-axis at just one point?

Answer: 4

With c = 4: x² − 4x + 4 = (x − 2)(x − 2). Both zeroes are 2, so it touches the x-axis only at (2, 0).

At most how many zeroes can a quadratic polynomial have?

Answer: 2

Two distinct zeroes, two equal zeroes (one zero), or none. The most is 2.

• none — correct. Yes! It never meets the x-axis.
• one. One zero means the curve touches the x-axis at one point. This one never touches.
• two. Two zeroes mean two meeting points. This curve never meets the axis.
• Two equal zeroes, so just one zero — correct. Yes! The two meeting points have joined into one.
• Two different zeroes. Different zeroes would mean two different meeting points.
• No zero. The curve does meet the x-axis, at one point.