How to Find Factored Form of a Parabola

To find the factored form of a parabola, you typically start with the standard quadratic equation, which is in the form of y = ax^2 + bx + c. The goal is to express this equation in the factored form, y = a(x – r1)(x – r2), where r1 and r2 are the roots of the quadratic equation.

Here’s a step-by-step method to achieve this:

1. Identify the coefficients: From your equation, note the values of a, b, and c.
2. Find the roots: You can use the quadratic formula, given by x = (-b ± √(b² – 4ac)) / (2a), to find the roots of the parabola. Calculate the discriminant (b² – 4ac) first to determine the number of real roots.
3. Plug the roots into the factored form: If you find real roots r1 and r2, substitute these values into the factored form. Your equation will now look like y = a(x – r1)(x – r2).

Example: Consider the quadratic equation y = 2x² – 4x – 6.

1. Here, a = 2, b = -4, and c = -6.
2. Using the quadratic formula, the roots can be calculated:
• Discriminant: (-4)² – 4(2)(-6) = 16 + 48 = 64
• Roots: x = (4 ± √64) / 4 = (4 ± 8) / 4
• This results in x = 3 and x = -1.
3. Thus, the factored form is y = 2(x – 3)(x + 1).

And that’s how you can find the factored form of a parabola!