How to convert vertex form to factored form?

Steps to Solve

1. Start with the vertex form of the equation

   We start with a quadratic in vertex form, given as ( y = a(x - h)^2 + k ). For example, let's assume ( a = 2 ), ( h = 3 ), and ( k = 1 ). The equation will be:

   $$ y = 2(x - 3)^2 + 1 $$

2. Expand the square

   Next, we expand the squared term ( (x - h)^2 ):

   $$ (x - 3)^2 = x^2 - 6x + 9 $$

   Now substitute this back into the equation:

   $$ y = 2(x^2 - 6x + 9) + 1 $$

3. Distribute the 'a' term

   We then distribute the ( a = 2 ) across the expanded expression:

   $$ y = 2x^2 - 12x + 18 + 1 $$

   Combine like terms:

   $$ y = 2x^2 - 12x + 19 $$

4. Factor the quadratic equation

   To factor the quadratic equation ( 2x^2 - 12x + 19 ), we first look for two numbers that multiply to ( 2 \times 19 = 38 ) and add to ( -12 ). In this case, the quadratic cannot be factored easily because it does not yield rational roots.

5. Using the quadratic formula (if necessary)

   If factoring is difficult, we can find the roots using the quadratic formula:

   $$ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

   For our equation, ( a = 2 ), ( b = -12 ), and ( c = 19 ):

   $$ r = \frac{-(-12) \pm \sqrt{(-12)^2 - 4 \cdot 2 \cdot 19}}{2 \cdot 2} $$ $$ r = \frac{12 \pm \sqrt{144 - 152}}{4} $$ $$ r = \frac{12 \pm \sqrt{-8}}{4} $$ Since the discriminant is negative, roots are complex.

6. Write in factored form

   Thus, since there are no real roots, we express the factored form in terms of complex numbers:

   $$ y = 2 \left( x - (3 + i\sqrt{2}) \right) \left( x - (3 - i\sqrt{2}) \right) $$