What does completing the square mean?
It means rewriting a quadratic so the x terms sit inside one squared bracket. You turn x² + bx into part of a perfect square trinomial like x² + 6x + 9, which equals (x + 3)².
The result is called vertex form. It shows the vertex of the parabola at a glance, and it lets you solve an equation by taking a square root instead of factoring.
The name comes from geometry. x² + 6x can be drawn as a square with two rectangles beside it, and the 9 you add is the small corner piece that fills in the full square.
Complete the square calculator example: x² + 6x + 5
The answer is (x + 3)² − 4. These are the lines the complete the square calculator walks through.
- Read the coefficients: a = 1, b = 6, c = 5.
- Halve b and square it: (6 ÷ 2)² = 3² = 9.
- Add and subtract 9: x² + 6x + 9 − 9 + 5.
- Group the perfect square: (x² + 6x + 9) − 4.
- Write it as a square: (x + 3)² − 4.
To check, expand it: (x + 3)² − 4 = x² + 6x + 9 − 4 = x² + 6x + 5. That matches the original, so nothing was lost.
Notice that 9 came from the x term alone. The original constant, 5, plays no part in picking the number. It just gets combined with −9 at the end.
What is the completing the square formula?
The general completing the square formula is ax² + bx + c = a(x + b/(2a))² + c − b²/(4a). In vertex form that means h = −b/(2a) and k = c − b²/(4a).
You rarely need to memorize this. Doing the steps with real numbers gives the same result, and the formula is mainly useful for checking h and k quickly.
How do you complete the square when a is not 1?
Factor a out of the x² and x terms first, then complete the square inside the bracket. Whatever you subtract inside gets multiplied by a when it comes out.
Take 2x² − 8x + 3. Pull out 2 from the first two terms to get 2(x² − 4x) + 3. Half of −4 is −2, and (−2)² = 4. So you write 2(x² − 4x + 4 − 4) + 3, which becomes 2(x − 2)² − 8 + 3, or 2(x − 2)² − 5.
That −8 is where most people slip. The −4 inside the bracket is multiplied by 2 on the way out.
How do you solve an equation by completing the square?
Complete the square, move the constant to the other side, and take the square root of both sides with ±. For x² + 4x + 1 = 0 the solutions are x = −2 ± √3.
This equation doesn't factor over whole numbers, which is the kind of problem where the method earns its keep.
What mistakes do students make when completing the square?
Adding the square without subtracting it is the most common one. If you add 9, you have changed the expression unless you also take 9 away (or add it to both sides of an equation).
Others include squaring b instead of half of b, forgetting the ± when you take a square root, and dropping the factor of a when a is not 1. Expanding your answer at the end catches all of these.
Why complete the square instead of factoring?
It works on every quadratic, including ones that do not factor nicely. It also gives you vertex form, which you need for graphing a parabola and for finding a maximum or minimum.
If you just want the roots and the numbers are friendly, factoring or the quadratic formula may be quicker.
How do you complete the square when a is negative?
Factor out the negative a, sign included, and carry on as usual. The minus sign stays in front of the bracket all the way to the end.
For −x² + 4x + 1, factor out −1 to get −(x² − 4x) + 1. Half of −4 is −2, and its square is 4. That gives −(x² − 4x + 4 − 4) + 1. When the −4 comes out of the bracket it is multiplied by −1, so it turns into +4: −(x − 2)² + 4 + 1 = −(x − 2)² + 5.
The same idea handles 3x² + 6x − 2. Factor out 3 to get 3(x² + 2x) − 2, add and subtract 1 inside, and the result is 3(x + 1)² − 5. The calculator shows this factoring step every time a is anything other than 1.