Completing the Square
Write a quadratic as (x + p)² + q by halving the x coefficient and correcting the constant, then solve it by taking the square root of both sides.
What a learner can do afterwards
- Write x² + 6x + 1 as (x + 3)² - 8
- Solve from the completed square by taking the square root and keeping both signs
- Complete the square when the x² coefficient is not 1 by taking it out first
1 · Read
You have seen perfect square brackets before: (x + 5) squared expands to x squared + 10x + 25, and you can spot the pattern and take it apart. This stop goes the other way. It forces any quadratic into the shape (x + p) squared + q, even when the constant does not fit yet. That finished form is the key, because (x + p) squared + q = 0 can be solved by taking a square root, and the finished form (x + p) squared + q reads off the lowest point of the curve.
Here is why the moves work. In x squared + 6x, the x coefficient is 6. Half of 6 is 3, and 3 squared is 9. The perfect square (x + 3) squared expands to x squared + 6x + 9, so it matches the first two terms exactly if we add 9. But adding 9 changes the value of the expression, so we also subtract 9 to keep it the same. Add and subtract the same number and nothing has really changed, but now the first three terms form a perfect square.
Write x squared + 6x + 1 as (x + p) squared + q. Halve the x coefficient: half of 6 is 3. Square it: 9. Add and subtract 9 in the middle: x squared + 6x + 9 - 9 + 1. The first three terms are (x + 3) squared, so the expression becomes (x + 3) squared - 8. Check by expanding: (x + 3) squared is x squared + 6x + 9, and 9 - 8 is 1, so we are back to x squared + 6x + 1.
Now use the finished form to solve. Solve x squared + 6x + 1 = 0. We just wrote the left side as (x + 3) squared - 8, so the equation is (x + 3) squared - 8 = 0, which rearranges to (x + 3) squared = 8. Take the square root of both sides and keep both signs: x + 3 = the square root of 8, or x + 3 = minus the square root of 8. So x = -3 plus or minus the square root of 8.
Now the x squared term does not start at 1. Write 2x squared + 8x + 3 in the form a(x + p) squared + q. Take out the factor 2 from the first two terms: 2(x squared + 4x) + 3. Inside the bracket, halve the x coefficient: half of 4 is 2, and 2 squared is 4. Add and subtract 4 inside: 2(x squared + 4x + 4 - 4) + 3. The bracket is (x + 2) squared, so this is 2((x + 2) squared - 4) + 3, which expands to 2(x + 2) squared - 8 + 3, that is 2(x + 2) squared - 5.
The finished form reads off the lowest point of the curve as well. In (x + p) squared + q, the square part is never negative, so the smallest value the expression can take is q, and it happens where the square is zero, that is at x = -p. For (x + 3) squared - 8 the minimum is -8 at x = -3, and for 2(x + 2) squared - 5 it is -5 at x = -2.
Halve the x coefficient, square it, add and subtract that number to build a perfect square, then solve by taking the square root with both signs.
2 · Watch
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Where this leads
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24 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.