Part 28 of 30 · QuadraticsSee the whole course
Factored form of a quadratic: reading the roots
In factored form, y = a(x − p)(x − q), the roots are x = p and x = q. They are where the parabola lands on the x-axis. A product is zero only when one of its brackets is zero, and that is why the roots can be read straight off.
What a root is
A root of a quadratic, also called a zero or an x-intercept, is an x that makes y equal to 0. On the graph, it is a spot where the parabola crosses the x-axis. A quadratic can have two roots, one, or none.
The zero product property
If two numbers multiply to give 0, at least one of them must be 0. So in (x − 1)(x − 5) = 0, either x − 1 = 0 or x − 5 = 0. That gives x = 1 or x = 5. This works only when the other side is exactly 0.
Example 1: y = (x − 1)(x − 5)
Roots: x − 1 = 0 gives x = 1, and x − 5 = 0 gives x = 5.
The vertex is halfway between the roots, at x = (1 + 5) ÷ 2 = 3. The height there is (3 − 1)(3 − 5) = 2 × (−2) = −4, so the vertex is (3, −4).
Expanded, it is x² − 6x + 5.
The figure below starts with this one. Move the roots and watch the vertex stay in the middle.
- (x − 1) = 0 at x = 1, and (x − 5) = 0 at x = 5
- vertex halfway between the roots: x = (1 + 5) ÷ 2 = 3
- vertex height: −4, so the vertex is (3, −4)
Example 2: mind the plus sign
y = (x + 2)(x − 4). The bracket (x + 2) is zero when x = −2, not 2.
The roots are −2 and 4. The vertex is at x = (−2 + 4) ÷ 2 = 1, with height (3)(−3) = −9, so (1, −9).
Example 3: a missing bracket and a double root
y = x(x − 6): x is zero at x = 0, and x − 6 is zero at x = 6. The roots are 0 and 6, and the vertex is (3, −9).
y = (x − 3)²: both brackets are the same, so there is one root, x = 3. The parabola just touches the x-axis there.
What a number in front does
With y = −(x − 1)(x − 5) the roots are still 1 and 5, but the minus flips the parabola into an arch with the vertex at (3, 4). A number in front never changes the roots, only the height and direction. You can try it with the button in the figure.
Watch it explained in under a minute
The four mistakes to avoid
- Getting the sign wrong. (x + 2) gives the root −2.
- Finding only one root. Each bracket gives its own.
- Using the zero product property when the product is not 0. (x − 1)(x − 5) = 3 does not mean a bracket equals 3.
- Taking a number in front as a root. In −2(x − 4)(x + 4), the roots are 4 and −4, not −2.
Try it yourself
Tap a question to see the answer.
1. y = (x − 2)(x − 7): find the roots and the x of the vertex.
Roots 2 and 7. The vertex is halfway: x = (2 + 7) ÷ 2 = 4.5.
2. y = (x + 3)(x + 1): find the roots and the vertex.
Roots −3 and −1. The vertex is at x = −2, with height (1)(−1) = −1, so (−2, −1).
3. y = −2(x − 4)(x + 4): find the roots and the vertex.
Roots 4 and −4. The vertex is at x = 0, with height −2(−4)(4) = 32, so (0, 32), the top of an arch.
Where this fits in
Reading roots from factors is easy. The real work is getting the factors in the first place. That is next: solve a quadratic by factoring.