Put the quadratic into vertex form and state the coordinates of the vertex
y=x^2-12x+45
Vertex form:y=___________
Vertex:(,)
To put the quadratic equation y=x^2-12x+45 into vertex form, we need to complete the square. The general form of a quadratic equation in vertex form is y=a(x-h)^2+k, where (h,k) represents the coordinates of the vertex.
First, we need to factor out the coefficient of x^2, which is 1:
y=(x^2-12x)+45
To complete the square, we need to take half of the coefficient of x (-12/2=-6) and square it (-6)^2=36. We add this value inside the parentheses and subtract the same value from outside the parentheses to keep the equation balanced:
y=(x^2-12x+36-36)+45
Simplifying, we have:
y=(x^2-12x+36)-36+45
Notice that the expression inside the parentheses, x^2-12x+36, can be factored as a perfect square: (x-6)^2. Now we can rewrite the equation as:
y=(x-6)^2+9
The quadratic equation y=x^2-12x+45 can be written in vertex form as y=(x-6)^2+9. The coordinates of the vertex are (6, 9).