Question

Convert y = 9x2 + 108x - 72 to vertex form by completing the square.

Answer 1

Expressing the equation in vertex form we have;

`y=9(x+6)^2-396`

Vertex at (-6,-396)

Step-by-step explanation:

We want to convert the quadratic equation given to vertex form by completing the square.

`y=9x^2+108x-72`

The vertex form of quadratic equation is of the form;

`f(x)=a(x-h)^2+k`

To do this by completing the square;

Firstly, let's add 72 to both sides of the qeuation;

`\begin{gathered} y+72=9x^2+108x-72+72 \\ y+72=9x^2+108x \end{gathered}`

Them we will add a number that can make the right side of the equation a complete square to both sides;

Adding 324 to both sides;

`\begin{gathered} y+72+324=9x^2+108x+324 \\ y+396=9x^2+108x+324 \end{gathered}`

factorizing the right side of the equation;

`\begin{gathered} y+396=9(x^2+12x+36) \\ y+396=9(x+6)(x+6) \\ y+396=9(x+6)^2 \end{gathered}`

Then, let us subtract 396 from both sides;

`\begin{gathered} y+396-396=9(x+6)^2-396 \\ y=9(x+6)^2-396 \end{gathered}`

Therefore, expressing the equation in vertex form we have;

`y=9(x+6)^2-396`

Vertex at (-6,-396)