Convert the following function into vertex form of the quadratic equation by completing the square. Show each step
y= 0.6(x^2-0.84)+0.21
To convert the given function into vertex form, we need to complete the square. The vertex form of a quadratic equation is given by y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
Let's start by expanding the expression inside the parenthesis:
y = 0.6(x^2 - 0.84) + 0.21
y = 0.6x^2 - 0.504 + 0.21
Combine the constant terms:
y = 0.6x^2 - 0.294
Now, let's complete the square. To do this, we need to take half of the coefficient of x (which is 0.6) and square it. Half of 0.6 is 0.3, and 0.3 squared is 0.09. Add this value inside the parenthesis and subtract it outside the parenthesis to maintain the equation's balance.
y = 0.6(x^2 - 0.294) [Add and subtract 0.09 inside the parenthesis]
Now, let's rewrite the equation by factoring the expression inside the parenthesis:
y = 0.6(x^2 - 0.294)
In order to make the expression inside the parenthesis a perfect square, we need to add (0.3)^2 = 0.09:
y = 0.6(x^2 - 0.294 + 0.09)
Simplify the expression inside the parenthesis:
y = 0.6(x^2 - 0.204)
Now, the expression inside the parenthesis can be written as a perfect square: (x - 0.3)^2.
To balance the equation, we also need to subtract the same value outside the parenthesis. Since we added 0.09 inside the parenthesis, we need to subtract 0.09 outside the parenthesis.
y = 0.6(x - 0.3)^2 - 0.09
So, the given function y = 0.6(x^2-0.84)+0.21, when converted to vertex form by completing the square, is y = 0.6(x - 0.3)^2 - 0.09.