Factored Form & Roots
You already know the standard form of a quadratic: y = ax² + bx + c. But there is another way to write the exact same function that makes the roots jump out at you. It is called factored form.
What Is Factored Form?
Instead of writing out all the terms, we write:
y = a(x - r₁)(x - r₂)
where r₁ and r₂ are the roots — the x-values where the parabola crosses the x-axis. Use the sliders below to see how each part works.
Try these experiments:
- Move r₁ and r₂ — the roots slide along the x-axis in real time.
- Set r₁ = r₂ — the parabola just touches the axis (a double root).
- Flip a to a negative value — the parabola opens downward, but the roots stay in the same spots!
Why Are They Called “Roots”?
Look at the equation y = a(x - r₁)(x - r₂). When you plug in x = r₁, the first factor becomes zero, so y = 0. Same thing for x = r₂. That is exactly why r₁ and r₂ are the x-intercepts — they make the whole expression zero.
Expanding to Standard Form
If you multiply out a(x - r₁)(x - r₂), you get standard form. Here is the algebra:
- First: (x - r₁)(x - r₂) = x² - (r₁ + r₂)x + r₁·r₂
- Then multiply by a: y = ax² - a(r₁ + r₂)x + a·r₁·r₂
So the coefficients are related:
- b = -a(r₁ + r₂)
- c = a · r₁ · r₂
Connecting the forms: The vertex form y = a(x - h)² + k tells you the peak or valley. The factored form y = a(x - r₁)(x - r₂) tells you the x-intercepts. The standard form y = ax² + bx + c tells you the y-intercept (when x = 0, y = c). All three describe the same parabola — they just highlight different features.
Comparing Factored Form vs Standard Form
Here is the same parabola drawn from both perspectives. They overlap perfectly because they are the same function written two different ways.
The two curves sit right on top of each other — factored form and standard form are just two lenses for viewing the same quadratic.
Challenge: Can you set the sliders so that the parabola passes through (−1, 0) and (4, 0) and has its vertex at y = −6.25? Hint: think about what a must be once you know the roots.