How do you convert vertex form to factored form $ y = 3{(x + 7)^2} - 2 $ ?

Hint:The question is in the form of vertex form that is $ y = a{(x - h)^2} + k $ where $ (h,k) $ is the vertex. This form of equation is converted into factored form where factored term is the product of a constant and two linear terms and they are the roots of the function.

Complete step by step explanation:
Here the given equation is in the form of vertex form and we have to convert this given into the factored form. The vertex form of equation is defined as $ y = a{(x
- h)^2} + k $ where $ (h,k) $ is the vertex. The factored form can also be called a standard quadratic form. By the factored form we can also determine the roots of the equation. The factored term is defined as the product of a constant and two linear terms and they are the roots of the function.

Now consider the given equation $ y = 3{(x + 7)^2} - 2 $
Apply the standard algebraic formula $ {(a + b)^2} $ to $ {(x + 7)^2} $ . The standard formula $ {(a +
b)^2} $ is given as $ {(a + b)^2} = {a^2} + 2ab + {b^2} $
Then we have
 $ \Rightarrow y = 3({x^2} + 49 + 2 \times x \times 7) - 2 $
 $ \Rightarrow y = 3({x^2} + 49 + 14x) - 2 $
On multiplying,
 $ \Rightarrow y = 3{x^2} + 147 + 42x - 2 $
 $ \Rightarrow y = 3{x^2} + 42x + 145 $
The above equation is in the form of quadratic equation and by using formula $ \dfrac{{ - b \pm
\sqrt {{b^2} - 4ac} }}{{2a}} $ , we can find the factors for the equation.
To the equation $ y = 3{x^2} + 42x + 145 $ we determine the roots for this equation by using the
formula $ \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $ , where a = 3, b = 42 and c = 145.
Therefore, we have
 $ x = \dfrac{{ - 42 \pm \sqrt {{{(42)}^2} - 4(3)(145)} }}{{2(3)}} $

 $
\Rightarrow x = \dfrac{{ - 42 \pm \sqrt {1764 - 1740} }}{6} \\
\Rightarrow x = \dfrac{{ - 42 \pm \sqrt {24} }}{6} \\
 $
The $ \sqrt {24} $ can be written as $ \sqrt {6 \times 4} = \sqrt 6 \sqrt 4 = 2\sqrt 6 $
So we have
 $
\Rightarrow x = \dfrac{{ - 42 \pm 2\sqrt 6 }}{6} \\
\Rightarrow x = \dfrac{{ - 42}}{6} \pm \dfrac{{2\sqrt 6 }}{6} \\

 $
On further simplification we have
 $ \Rightarrow x = - 7 \pm \dfrac{{\sqrt 6 }}{3} $
Therefore, the roots are $ x = - 7 + \dfrac{{\sqrt 6 }}{3} $ and $ x = - 7 - \dfrac{{\sqrt 6 }}{3} $

Therefore, the factored form of the equation $ y = 3{(x + 7)^2} - 2 $ is written as $ y = \left( {x + 7 -
\dfrac{{\sqrt 6 }}{3}} \right)\left( {x + 7 + \dfrac{{\sqrt 6 }}{3}} \right) $

Note: The vertex form and the factored form are the forms representing the equation. It can be quadratic form or any other form. We can convert these forms of equations by using the simple methods and by using the formulas. The vertex form converted into factored form by using the formula which is used to find roots of the equation.