Completing the square method

 Completing the square method

Completing the square is a mathematical technique used to rewrite a quadratic expression in a specific form called "vertex form" or "standard form." It is particularly useful when solving quadratic equations or graphing quadratic functions.

The general form of a quadratic expression is:

ax^2 + bx + c

To complete the square for this quadratic expression, the goal is to rewrite it in the form:

a(x - h)^2 + k

where (h, k) represents the vertex of the parabola defined by the quadratic expression.

Process:

Here are the steps to complete the square:

1. Make sure the coefficient of the x^2 term, 'a,' is equal to 1. If 'a' is not equal to 1, divide the entire expression by 'a' to set it to 1.

2. Group the x-terms together and leave the constant term, 'c,' on the other side of the equation.

   ax^2 + bx = -c

3. Take half of the coefficient of the x-term, 'b,' and square it. This value will be used to complete the square. Let's call it (b/2)^2.

   (b/2)^2

4. Add the value from step 3 to both sides of the equation.

   ax^2 + bx + (b/2)^2 = -c + (b/2)^2

   This step ensures that the left side of the equation can be factored into a perfect square.

5. Factor the left side of the equation as a perfect square.

   (x + b/2)^2 = -c + (b/2)^2

6. Simplify the right side of the equation, if necessary.

7. Take the square root of both sides of the equation.

   x + b/2 = ±√(-c + (b/2)^2)

8. Solve for 'x' by subtracting b/2 from both sides of the equation.

   x = -b/2 ± √(-c + (b/2)^2)

The quadratic expression has now been rewritten in vertex form, allowing you to easily identify the coordinates of the vertex (h, k) as (-b/2, -c + (b/2)^2).

Completing the square is a useful technique because it provides insights into the shape and location of the graph of a quadratic function. It can also be used to solve quadratic equations by setting the expression equal to zero and solving for 'x' using the derived formula