2.2 Completing The Square

Completing the square is a technique used to rewrite a quadratic expression in the form a(x + p)² + q.

This form is particularly useful for finding the vertex of a parabola, determining minimum or maximum values, and solving quadratic equations.

The technique is based on the algebraic identity (x + p)² = x² + 2px + p².

To complete the square for a quadratic expression in the form x² + bx + c:

1. Take half the coefficient of x and square it: (b/2)²

2. Add and subtract this value to maintain equality: x² + bx + (b/2)² - (b/2)² + c

3. Rewrite the first three terms as a perfect square: (x + b/2)² - (b/2)² + c

4. Simplify: (x + b/2)² + (c - (b/2)²)

When the coefficient of x² is not 1, we need to factor it out first:

1. Factor out a from the first two terms: a(x² + (b/a)x) + c

2. Complete the square inside the parentheses: a((x + (b/2a))² - (b/2a)²) + c

3. Simplify: a(x + (b/2a))² - a(b/2a)² + c = a(x + (b/2a))² + (c - b²/4a)

Completing the square can be used to solve quadratic equations:

1. Rewrite the equation in the form ax² + bx + c = 0

2. Complete the square to get a(x + p)² + q = 0

3. Solve for (x + p)² by subtracting q from both sides and dividing by a

4. Take the square root of both sides

5. Solve for x

The vertex of a quadratic function f(x) = ax² + bx + c is the point where the function reaches its minimum (if a > 0) or maximum (if a < 0) value.

By completing the square to write f(x) = a(x - h)² + k, the vertex is at the point (h, k).

Alternatively, the x-coordinate of the vertex is given by x = -b/(2a), and the y-coordinate can be found by substituting this value into the function.

When a quadratic function is written in the form f(x) = a(x - h)² + k, the value k is the minimum value of the function if a > 0, or the maximum value if a < 0.

This is because (x - h)² is always non-negative, and its minimum value is 0 when x = h.

Therefore, the minimum or maximum value of f(x) occurs at the vertex (h, k).