Completing The Square

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Completing the Square

If a quadratic equation is of the form (x + h)2 = d, then it could be solved by finding the square roots of both sides.  This section of Chapter 6 introduces a method so we can write a quadratic equation in this form.

Recall the relationship between the constant in the binomial and the coefficients that occur in a trinomial square.  For example:

This relationship suggests a process for changing a trinomial to a perfect square trinomial.  The process is called completing the square.

example #1:
Complete the square.



x2 – 8x + 16  =  (x – 4)2

Solving Quadratic Equations

We can use the technique of completing the square to solve quadratic equations.  Remember that the addition property allows us to add a number to both sides of an equation.

example #2:
Solve by completing the square.



The solutions are and .

To use the technique of completing the square to solve a quadratic equation, the coefficient of the leading term must be 1.  If the leading coefficient is not 1, we can use the multiplication property to make it 1.

example #3:
Solve by completing the square.



The solutions are and .

You can use the following steps to complete the square and solve equations using the square root property.

  1. Divide each term by the leading coefficient if it is greater than 1.
  2. Using the addition property, move the constant term to the other side of the equation, isolating it from the variable terms.
  3. Using the coefficient on the linear term, take half of this number and square it.
  4. Add the resulting value to both sides of the equation.
  5. Factor the trinomial square to the form (x + h)2.
  6. Take the square root of both sides.
  7. Solve the equation.
  8. Simplify any radical expression completely.