To complete the square, you must follow these steps:
1. Place brackets around the quadratic (x²) and linear (x) terms, while leaving the coefficient on the outside of the brackets.
2. Factor the terms in the inside of the brackets so that the quadratic term is equal to 1. It is important that you only factor out a coefficient. Do not factor out an x.
3. We are now going to work with the linear term. Take half of the linear term, square it, and add it and subtract it from the equation. When you add and subtract the same amount from an equation, you are essentially not changing the equation at all.
4. Keep the new added term inside the brackets; lets look at the subtracted term. Multiply it by the factor that you took out in Step 2, and move it outside of the brackets. Do not change the sign.
5. Inside the brackets, you now have a perfect square. Rearrange it.
6. Now, you can clearly see your vertex. The x coordinate can be found inside the brackets, and the y term can be found outside of the brackets. The a term can be found in front of the brackets.
Example 1: Complete the Square for the following expression: 4x² + 24x 1
4x² + 24x 1
= (4x² + 24x) 1 [place brackets around the quadratic and linear terms]
= 4 (x² + 6x) 1 [take out a common factor of 4 to make the x² term have a coefficient equal to 1]
= 4 (x² + 6x + 9 9) 1 [take half of 6, which is 3, and square it (3² = 9) and add it and subtract it from the expression]
= 4 (x² + 6x + 9) 1 36 [move the 9 out of the bracket, multiply it by 4, and keep the same sign]
= 4 (x² + 6x + 9) 37 [collect like terms]
= 4 (x + 3)² 37 [rearrange inside the brackets, to get a perfect square]
Thus, the vertex is at (-3,-37) [do not change the sign of the value inside the brackets].
The parabola opens up, because the a term is positive.
Example 2: Complete the Square for the following expression: -2x² 10x + 2
-2x² 10x + 2
= (-2x² 10x) + 2
= -2 (x² + 5x) + 2
= -2 (x² + 5x + (5/2)² (5/2)²) + 2
= -2 (x² + 5x + 25/4 25/4) +2
= -2 (x² + 5x + 25/4) + 2 + 25/2
= -2 (x² + 5x + 25/4) + 75/2
= -2 (x + 5/2)² + 75/2
Thus, the vertex is at the point (-5/2,75/2), and the parabola opens down.
Notice that when making a perfect square, the second term is always the square root of the last term inside the brackets.
Practice Questions
Page 390 #1de, 3c, 4 (column 2), 8aceg, 11
