Answer: To convert a quadratic equation from standard form (also known as general form) to vertex form, you need to complete the square. The vertex form of a quadratic equation is given by:
y = a(x - h)^2 + k,
where (h, k) represents the coordinates of the vertex, and 'a' is a constant representing the stretch or compression of the parabola.
Let's convert the given equation from standard form (0 = 5x^2 - 15x + 1) to vertex form (y = a(x - h)^2 + k):
Step 1: Factor out the common coefficient (if not already done).
0 = 5(x^2 - 3x) + 1.
Step 2: Complete the square inside the parentheses:
To complete the square for x^2 - 3x, we need to add and subtract the square of half of the coefficient of x, which is (3/2)^2 = 9/4.
0 = 5(x^2 - 3x + 9/4 - 9/4) + 1.
Step 3: Group the square term with the constant term:
0 = 5[(x - 3/2)^2 - 9/4] + 1.
Step 4: Distribute the common coefficient back:
0 = 5(x - 3/2)^2 - 45/4 + 1.
Step 5: Simplify the constant terms:
0 = 5(x - 3/2)^2 - 41/4.
Now, we have the equation in vertex form: y = 5(x - 3/2)^2 - 41/4.
The vertex of the parabola is (3/2, -41/4). The 'a' value is 5, representing that the parabola is stretched vertically.