Completing the Square

Completing the square is a technique for solving quadratic equations. By transforming the equation into a perfect square trinomial, it becomes easier to solve for \( x \). A quadratic equation generally has the form:

$$ ax^2 + bx + c = 0 $$

To complete the square, we want to rewrite the equation in the form \( (x + p)^2 = q \), where \( p \) and \( q \) are constants. This allows us to solve for \( x \) by taking the square root of both sides.

Steps for Completing the Square

  1. Ensure the coefficient of \( x^2 \) is 1. If it isn’t, divide both sides of the equation by \( a \).
  2. Move the constant term to the right side of the equation.
  3. Add \( \left(\frac{b}{2}\right)^2 \) to both sides to form a perfect square trinomial on the left side.
  4. Rewrite the left side as a squared term, \( (x + p)^2 \), and simplify the right side.
  5. Solve for \( x \) by taking the square root of both sides and isolating \( x \).

Example: Solving by Completing the Square

Consider the equation:

$$ x^2 + 6x - 7 = 0 $$

Step-by-step solution:

  1. Move the constant term to the other side:

    2. $$ x^2 + 6x = 7 $$

  2. Find \( \left(\frac{6}{2}\right)^2 = 9 \) and add it to both sides:

    3. $$ x^2 + 6x + 9 = 7 + 9 $$

    So the equation becomes:

    $$ (x + 3)^2 = 16 $$

  3. Take the square root of both sides:

    4. $$ x + 3 = \pm 4 $$

  4. Isolate \( x \):
    • \( x = -3 + 4 = 1 \)
    • \( x = -3 - 4 = -7 \)

    The solutions are \( x = 1 \) and \( x = -7 \).

Practice Questions

  1. Solve \( x^2 + 4x - 5 = 0 \) by completing the square.
    Solution

    Move the constant term to the other side:

    $$ x^2 + 4x = 5 $$

    Find \( \left(\frac{4}{2}\right)^2 = 4 \) and add it to both sides:

    $$ x^2 + 4x + 4 = 5 + 4 $$

    Rewrite as a squared term:

    $$ (x + 2)^2 = 9 $$

    Take the square root of both sides:

    • \( x + 2 = 3 \Rightarrow x = 1 \)
    • \( x + 2 = -3 \Rightarrow x = -5 \)

    The solutions are \( x = 1 \) and \( x = -5 \).

  2. Solve \( x^2 - 10x + 16 = 0 \) by completing the square.
    Solution

    Move the constant term to the other side:

    $$ x^2 - 10x = -16 $$

    Find \( \left(\frac{-10}{2}\right)^2 = 25 \) and add it to both sides:

    $$ x^2 - 10x + 25 = -16 + 25 $$

    Rewrite as a squared term:

    $$ (x - 5)^2 = 9 $$

    Take the square root of both sides:

    • \( x - 5 = 3 \Rightarrow x = 8 \)
    • \( x - 5 = -3 \Rightarrow x = 2 \)

    The solutions are \( x = 8 \) and \( x = 2 \).

  3. Solve \( x^2 + 12x + 36 = 0 \) by completing the square.
    Solution

    This equation is already in the form of a perfect square.

    $$ (x + 6)^2 = 0 $$

    Take the square root of both sides:

    $$ x + 6 = 0 $$

    The solution is \( x = -6 \).

→The Nature of Roots and the Discriminant