Example 1
Question: Write the expression \(x^2 + 6x - 4\) in the form \((x+a)^2 + b\).
Show Solution
| \(x^2 + 6x - 4\) | Coefficient of \(x\) is 6 → half is 3. |
| \((x+3)^2 - 9 - 4\) | Write as complete square and balance constant. \(-3^2-4\). |
| \((x+3)^2 - 13\) | Simplify constants. |
| 💡 Final completed square form: \((x+3)^2 - 13\). | |
Example 2
Question: Complete the square for \(x^2 - 8x + 7\).
Show Solution
| \(x^2 - 8x + 7\) | Coefficient of \(x\) is 8 → half is 4. |
| \((x-4)^2 - 16 + 7\) | Rewrite with complete square. |
| \((x-4)^2 - 9\) | Simplify constants. |
| 💡 Final form is always a square term plus/minus a constant. | |
Example 3
Question: Complete the square for \(x^2 + 5x + 3\).
Show Solution
| \(x^2 + 5x + 3\) | Coefficient of \(x\) is 5 → half is \(\tfrac{5}{2}\). |
| \(\Big(x + \tfrac{5}{2}\Big)^2 - \Big(\tfrac{5}{2}\Big)^2 + 3\) | Write as complete square and balance constant. |
| \(-\tfrac{25}{4} + 3 = -\tfrac{13}{4}\) | Simplify constants. |
| \(x^2 + 5x + 3 = \Big(x + \tfrac{5}{2}\Big)^2 - \tfrac{13}{4}\) | Final completed square form. |
| 💡 The number inside the bracket is always half the coefficient of \(x\). | |
Example 4
Question: Complete the square for \(2x^2 - 6x - 5\).
Show Solution
| \(2x^2 - 6x - 5\) | Factor out coefficient of \(x^2\). |
| 2( \(x^2 - 3x\) \() - 5\) | Work inside the bracket. |
| \(x^2 - 3x = \big(x - \tfrac{3}{2}\big)^2 - \tfrac{9}{4}\) | Complete the square inside the bracket. |
| \(2\big(\)\(\big(x - \tfrac{3}{2}\big)^2 - \tfrac{9}{4}\)\(\big) - 5\) | Substitute the completed square back into the expression. |
| \(2\big(x - \tfrac{3}{2}\big)^2 - \tfrac{9}{2} - 5\) | Multiply out the constant: \(2\times\frac{9}{4}=\tfrac{9}{2}\). |
| \(2\big(x - \tfrac{3}{2}\big)^2 - \tfrac{19}{2}\) | Final completed square form. |
Example 5
Question: Complete the square for \(2x^2 - 12x + 3\).
Show Solution
| \(2x^2 - 12x + 3\) | Factor out coefficient of \(x^2\). |
| 2( \(x^2 - 6x\) ) + 3 | Work inside the bracket. |
| \(x^2 - 6x = (x - 3)^2 - 9\) | Complete the square inside the bracket. |
| \(2\big((x - 3)^2 - 9\big) + 3\) | Substitute back the completed square. |
| \(2(x - 3)^2 - 18 + 3\) | Expand and simplify constants. |
| \(2(x - 3)^2 - 15\) | Final completed square form. |
| 💡 Always factor out the coefficient of \(x^2\) first if it is not 1. | |
Example 6
Question: Express \(4x^2 + 20x + 5\) in the form \((ax+b)^2 + c\).
Show Solution
| \(4x^2 + 20x + 5\) | Notice \(4x^2 = (2x)^2\). Keep \(2x\) inside the bracket. |
| \((2x)^2 + 20x\) \(+ 5\) | Group first two terms to complete the square. |
| \((2x+5)^2 - 25\) \(+ 5\) | Half of 20 (relative to \(2x\)) is 5, so square gives 25. Balance by subtracting 25. |
| \((2x+5)^2 - 20\) | Simplify constants: \(-25 + 5 = -20\). |
| \(4x^2 + 20x + 5 = (2x+5)^2 - 20\) | Final completed square form. |
| 💡 Here we did not factor out 4. Instead, we wrote \(4x^2\) as \((2x)^2\) and completed the square directly. | |