A hidden root is a real solution that the current window does not show. It happens when a root lies outside the window, or when two roots are so close together that the curve looks like it only touches the axis. This lesson builds on choosing a viewing window.
How can a window hide a root?
There are two ways. The first is distance: the root is off the left, right, top or bottom of the screen. The second is scale: the window is so wide that two nearby roots merge into one point.
You cannot fix this by staring harder at the picture. You test it with algebra that does not depend on the screen.
Two quick algebra tests
For y = x² + bx + c with roots p and q:
- Sum of roots: p + q = −b.
- Product of roots: p × q = c.
- Discriminant: b² − 4c. If it is positive there are two roots, zero means one repeated root, and negative means none.
If the screen shows one root p, then the other is q = −b − p. You can check it by substitution.
Worked example
The calculator shows y = x² − 52x + 100 crossing the x-axis only at x = 2 in the standard window. Is there another root?
Step 1, sum of roots: b = −52, so p + q = 52. With p = 2, q = 50.
Step 2, product check: 2 × 50 = 100, which equals c. Both tests agree.
Step 3, confirm by factorising: x² − 52x + 100 = (x − 2)(x − 50), so the roots are 2 and 50.
Step 4, confirm on the calculator: set Xmin = −5, Xmax = 60, Ymin = −700, Ymax = 150. The turning point is (26, 676 − 1352 + 100) = (26, −576), so Ymin = −700 fits. Both crossings are now visible.
The mistake to watch for
Now take y = x² − 4x + 3.99 in a window of −10 to 10.
Mistaken conclusion: “The graph touches the x-axis at x = 2, so there is one repeated root.”
At this scale the curve dips only 0.01 below the axis, which is less than one pixel.
The correction is to calculate the discriminant: 16 − 4 × 3.99 = 16 − 15.96 = 0.04. It is positive, so there are two roots. They are x = (4 ± 0.2)/2, which is 1.9 and 2.1.
Check: 1.9 + 2.1 = 4 and 1.9 × 2.1 = 3.99. Zooming in around x = 2 with Ymin = −0.05 and Ymax = 0.05 would confirm this on screen.
Check yourself
1. The standard window shows y = x² − 33x + 32 crossing at x = 1 only. Find the other root.
Show answer
Sum of roots = 33, so the other root is 33 − 1 = 32. Product: 1 × 32 = 32 = c. The roots are 1 and 32.
2. y = x² − 6x + 8.91 seems to touch the axis near x = 3. How many roots does it have, and what are they?
Show answer
Discriminant = 36 − 35.64 = 0.36, which is positive, so there are two roots. x = (6 ± 0.6)/2, so x = 2.7 and x = 3.3. Check: 2.7 × 3.3 = 8.91 and 2.7 + 3.3 = 6.
3. A student widens the window again and again but never finds a root of y = x² + 1. What should they do instead?
Show answer
Calculate the discriminant: 0 − 4 = −4, which is negative, so there are no real roots. The minimum value is 1 at x = 0, so the curve stays above the x-axis and no window can show a crossing.
Where this leads next
Once you can tell a hidden root from a missing one, compare what the screen gives with exact working in graphical and exact solutions, then practise in the mixed practice set. Back to the module overview if you want the full route.
If you keep doubting what the graph shows you, a teacher in online one-to-one Mathematics tuition can build the habit of testing before trusting.