The roots of a quadratic equation are the x-coordinates of the points where its graph meets the x-axis. At those points y = 0, so they solve ax² + bx + c = 0.
This lesson belongs to SPM Mathematics quadratic functions and equations. It connects the graph to the algebra of solving factorisable quadratic equations.
What does the table of values tell us?
Here is an original function, y = x² − 4x − 5. Substituting whole numbers gives this table.
| x | −2 | −1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|---|---|---|
| y | 7 | 0 | −5 | −8 | −9 | −8 | −5 | 0 | 7 |
Plot the points and draw a smooth curve. It is a ∪ shape because a = 1 is positive.
Question 1: what are the roots of x² − 4x − 5 = 0?
Find where y = 0. The table shows y = 0 at x = −1 and x = 5, so the roots are x = −1 and x = 5.
You can confirm this with algebra: x² − 4x − 5 = (x + 1)(x − 5), which gives the same two values. A graph answer and an algebra answer should always agree.
Question 2: what are the turning point and axis of symmetry?
The lowest point is (2, −9). The axis of symmetry is the vertical line x = 2, which sits exactly halfway between the roots −1 and 5.
The midpoint shortcut is worth remembering: (−1 + 5) ÷ 2 = 2. It also explains why the table of values is symmetric on either side of x = 2.
Question 3: how do you solve x² − 4x − 5 = −5?
Draw the horizontal line y = −5. It cuts the curve at x = 0 and x = 4, so the solutions are x = 0 and x = 4.
Check in the equation: x² − 4x = 0 gives x(x − 4) = 0, which agrees. The horizontal line does the job that the x-axis does for a zero equation.
The mistake that costs marks
A student asked for the roots reads the y-intercept, −5, and writes “the root is −5”. The y-intercept is where x = 0, so it tells you about the value of c, not the roots.
Another slip is reading the y-coordinate of the turning point when asked for the axis of symmetry. The axis is x = 2, a vertical line, and −9 is the minimum value of y.
| Question asks for | Read from | Answer here |
|---|---|---|
| Roots of the equation = 0 | x-axis crossings (x-values) | −1 and 5 |
| Axis of symmetry | x-value of the turning point | x = 2 |
| Minimum value | y-value of the turning point | −9 |
| Solve equal to −5 | Line y = −5, read x-values | 0 and 4 |
Check yourself
The table for y = x² − 6x + 5 gives y = 12 at x = −1, 5 at x = 0, 0 at x = 1, −3 at x = 2, −4 at x = 3, −3 at x = 4, 0 at x = 5 and 5 at x = 6. Find the roots of x² − 6x + 5 = 0 and the solutions of x² − 6x + 5 = −3.
Answer
y = 0 at x = 1 and x = 5, so the roots are x = 1 and x = 5.
For the second equation, draw y = −3. The table gives y = −3 at x = 2 and x = 4, so the solutions are x = 2 and x = 4.
Algebra check: x² − 6x + 8 = 0 factorises as (x − 2)(x − 4) = 0.
What to study next
The lowest or highest point of the curve often carries the meaning in a word problem. Continue with interpreting a turning point in a contextual problem, and experiment with your own values in the quadratic graph and roots explorer.
If reading graphs is where you lose marks, online one-to-one Mathematics tuition lets a teacher check your sketches and labelling with you.