Question

fill in the blank 2.5 points find the vertex of the graph by completing the square. $x^2 + 12x - 45 = 0$ write answers as (x,y) with no spaces. type your answer...

Explanation:

Step1: Rewrite the equation

We have the quadratic equation \(y = x^{2}+12x - 45\) (we can think of it as a function to find the vertex of its graph). First, move the constant term to the other side: \(x^{2}+12x=45\).

Step2: Complete the square

To complete the square for the \(x\) terms, we take half of the coefficient of \(x\), which is \(\frac{12}{2} = 6\), and square it: \(6^{2}=36\). Add this to both sides of the equation: \(x^{2}+12x + 36=45 + 36\).

Step3: Rewrite as a perfect square

The left - hand side can be written as a perfect square: \((x + 6)^{2}=81\). So the equation of the parabola in vertex form \(y=(x - h)^{2}+k\) (where \((h,k)\) is the vertex) is \(y=(x + 6)^{2}-81\) (since \(y=(x + 6)^{2}-81\) is equivalent to \(y=x^{2}+12x+36 - 81=x^{2}+12x - 45\)). In the vertex form \(y=(x - (- 6))^{2}+(-81)\), the vertex \((h,k)\) is \((-6,-81)\).

Answer:

\((-6,-81)\)