Solving Quadratic Equations by Completing the Square
 Topic Index | Algebra2/Trig Index | Regents Exam Prep Center

 

Solving quadratic equations by completing the square is overpowered by an "offspring" of this process, namely, the quadratic formula.  The quadratic formula was derived by completing the square on a quadratic equation.  Once  the quadratic formula was derived, it was no longer necessary to use the process of completing the square to solve "each" quadratic equation.  Even though completing the square is often overlooked in favor of the quadratic formula, it is still a valuable skill that will be needed in other mathematical situations.  Therefore, it is worthwhile to "get our feet wet" with these easier examples of applying the process of completing the square.

The process of completing the square was explained in the section entitled Complete the Square.

Examples:

1. 

Keep all terms containing x on one side.  Move the constant to the right.

Get ready to create a perfect square on the left.  Balance the equation.

Take half of the x-term coefficient and square it.  Add this value to both sides.

Simplify and write the perfect square on the left.

Take the square root of both sides.  Be sure to allow for both plus and minus.

Solve for x.

 

2. 

Keep all terms containing x on one side.  Move the constant to the right.

Get ready to create a perfect square on the left.  Balance the equation.

Take half of the x-term coefficient and square it.  Add this value to both sides.

Simplify and write the perfect square on the left.

Take the square root of both sides.  Be sure to allow for both plus and minus.

Represent the negative radical as an imaginary number and solve for x.



3. 

Keep all terms containing x on one side.  This equation is all set up to start.

Divide all terms by 5 to create a leading coefficient of one.

Prepare to get a perfect square on the left.  Balance the equation.

Take half of the x-term coefficient and square it.  Add this value to both sides.

Simplify

Simplify


Write the perfect square on the left.

Take the square root of both sides.  Be sure to allow for both plus and minus.

Solve for x.