Question

4 convert the equation $y = (x - 7)^2 - 13$ to factored form and type the factored form in the box below. the factored form equation is $y = $

Explanation:

Step1: Recall the difference of cubes formula

The difference of cubes formula is \(a^3 - b^3=(a - b)(a^2+ab + b^2)\). Here, we have \(y=(x - 7)^3-13\), so we can let \(a=(x - 7)\) and \(b=\sqrt[3]{13}\) (since \(13 = (\sqrt[3]{13})^3\)).

Step2: Apply the difference of cubes formula

Substitute \(a=(x - 7)\) and \(b = \sqrt[3]{13}\) into the difference of cubes formula:
\[

$$\begin{align*} y&=(x - 7)^3-(\sqrt[3]{13})^3\\ &=(x - 7-\sqrt[3]{13})((x - 7)^2+(x - 7)\sqrt[3]{13}+(\sqrt[3]{13})^2) \end{align*}$$

\]
We can also simplify \((x - 7)^2=x^2-14x + 49\) and \((\sqrt[3]{13})^2=\sqrt[3]{169}\), so the second factor becomes \(x^2-14x + 49+(x - 7)\sqrt[3]{13}+\sqrt[3]{169}\)

Answer:

\(y=(x - 7-\sqrt[3]{13})(x^2-14x + 49+(x - 7)\sqrt[3]{13}+\sqrt[3]{169})\)