Part 35 of 36 · QuadraticsSee the whole course
Completing the square, step by step
Completing the square rewrites x² + bx + c as (x + b/2)² + (c − b²/4). You add the square of half of b to make a perfect square, and take the same amount away so the value does not change. The result is vertex form, so you can read the vertex, and you can solve the equation by taking a square root.
The picture: a square with a corner missing
Look at x² + 6x. Draw a square with side x, which has area x². Then add two strips, each 3 wide and x long, which have area 3x each, 6x in all. You almost have a bigger square, but there is a corner missing: a 3 by 3 square, with area 9.
So add 9 to complete the square. But you cannot just add 9 and walk away: that changes the number. Add 9 and take 9 away, and the value stays the same.
The steps (when the number in front of x² is 1)
- Halve b, the number in front of x. For x² + 6x + 5, half of 6 is 3.
- Square it. 3² = 9.
- Add and subtract it: x² + 6x + 9 − 9 + 5.
- Write the first three terms as a square: (x + 3)² − 9 + 5.
- Tidy up the numbers: (x + 3)² − 4.
Try it. Move b and c. The vertex is always at x = −b/2, and the number that is left over, c − b²/4, is its height.
- half of 6 is 3, and 3² = 9
- y = (x² + 6x + 9) − 9 + 5
- y = (x + 3)² − 4
Example 1: find the vertex of y = x² + 6x + 5
Half of 6 is 3, and 3² = 9
y = (x² + 6x + 9) − 9 + 5
y = (x + 3)² − 4
This is vertex form, y = (x − h)² + k, with h = −3 and k = −4. The vertex is (−3, −4).
Example 2: solve x² + 6x + 5 = 0
Use the same completed square, then undo it with a square root.
(x + 3)² − 4 = 0
(x + 3)² = 4
x + 3 = 2 or x + 3 = −2 (both signs: 2² and (−2)² are 4)
x = −1 or x = −5
Check: (−1)² + 6(−1) + 5 = 1 − 6 + 5 = 0 ✓ and (−5)² + 6(−5) + 5 = 25 − 30 + 5 = 0 ✓
This one also factors, as (x + 1)(x + 5), which is the same answer by factoring. The point of completing the square is that it works even when factoring does not.
Example 3: it does not factor, x² − 8x + 3 = 0
Half of −8 is −4, and (−4)² = 16
Move the 3 across: x² − 8x = −3
Add 16 to both sides: x² − 8x + 16 = 13
(x − 4)² = 13
x − 4 = √13 or x − 4 = −√13
x = 4 + √13 or x = 4 − √13, about 7.61 or 0.39
Check x = 4 + √13: x² = 29 + 8√13, and −8x = −32 − 8√13, so 29 − 32 + 3 = 0 ✓
Notice the sign: half of −8 is −4, so the square is (x − 4)². The sign inside the bracket is the sign of half of b.
Example 4: when the number in front of x² is not 1, y = 2x² + 12x + 5
First take the 2 out of the x-terms. Then complete the square inside the bracket.
y = 2(x² + 6x) + 5
Inside the bracket, add and subtract 9: y = 2(x² + 6x + 9 − 9) + 5
Take the −9 out of the bracket. It is multiplied by 2, so it becomes −18: y = 2(x + 3)² − 18 + 5
y = 2(x + 3)² − 13
The vertex is (−3, −13).
That −9 is the trap. It sits inside a bracket that is multiplied by 2, so what comes out is 2 × (−9) = −18.
Where the quadratic formula comes from
Complete the square on the general equation ax² + bx + c = 0 and you get x = (−b ± √(b² − 4ac)) ÷ 2a. The quadratic formula is just this method done once, for every quadratic at the same time. See the quadratic formula and the discriminant.
Watch it explained in under a minute
Read the video transcript
Here is a quadratic: x² + 6x + 5. We want a form that shows the vertex. The trick: completing the square.
Take half of the 6, which is 3. Square it: 9. Now add 9, and take 9 away.
The first three terms are now a perfect square. (x + 3)². And −9 + 5 is −4.
This is vertex form. The vertex is at (−3, −4). The graph agrees.
To solve, set it equal to 0. (x + 3)² = 4. So x + 3 is 2, or −2. x = −1, or x = −5.
Want practice questions and more examples? Full guide: pipmath.app/blog
The five mistakes to avoid
- Squaring b instead of half of b. For x² + 6x, add 3² = 9, not 6² = 36.
- Adding the square without taking it away. That changes the equation. (In an equation, add it to both sides.)
- Writing the wrong sign in the bracket. x² − 8x + 16 is (x − 4)², not (x + 4)².
- Forgetting the ± when you take the square root. A square root gives two answers.
- With a number in front of x², forgetting to multiply the amount you take out by that number.
Try it yourself
Tap a question to see the answer.
1. Complete the square: x² + 10x + 7
Half of 10 is 5, and 5² = 25. x² + 10x + 25 − 25 + 7 = (x + 5)² − 18. The vertex is (−5, −18).
2. Solve x² − 4x − 5 = 0 by completing the square
Half of −4 is −2, and (−2)² = 4. Add 5 to both sides: x² − 4x = 5. Add 4: (x − 2)² = 9. So x − 2 = 3 or x − 2 = −3, which gives x = 5 or x = −1. Check: 25 − 20 − 5 = 0 ✓.
3. Find the vertex of y = x² − 6x + 4
Half of −6 is −3, and (−3)² = 9. y = (x² − 6x + 9) − 9 + 4 = (x − 3)² − 5. The vertex is (3, −5).
Where this fits in
You now have three ways to solve a quadratic: factoring when the numbers are friendly, completing the square when you want the vertex or a clean method, and one formula that solves them all. That formula is next: the quadratic formula and the discriminant.