Final answer:
To express the equation y = x^3 - 7x^2 + 10x in vertex form, we need to complete the square.
The vertex form of the given equation is y = (x + 5)^2(x - 7).
Step-by-step explanation:
Step by step explanation:
First, let's group the terms involving 'x' together:
y = (x^3 - 7x^2) + 10x.
Next, factor out the greatest common factor from the terms within the parentheses:
y = x^2(x - 7) + 10x.
Now, we have a quadratic expression inside the parentheses.
To complete the square, we need to add and subtract the square of half of the coefficient of 'x':
y = x^2(x - 7) + 10x + (-3.5)^2 - (-3.5)^2.
Simplify: y = x^2(x - 7) + 10x + 12.25 - 12.25.
Factor the quadratic expression: y = (x^2 + 10x + 12.25)(x - 7).
Finally, simplify further: y = (x + 5)^2(x - 7). This is the vertex form of the given equation.