Completing the Square to Solve a Quadratic
Completing the square turns any quadratic into (x + something)2 = number, which you solve with one square root. It works because every quadratic can be rewritten to show a perfect square hiding inside it.
A quadratic like x2 + bx + c is hard to solve directly because x shows up twice. Completing the square fixes that by rewriting the x2 and x terms as part of a perfect square binomial, so x only appears once, inside one set of parentheses. Once you have (x + h)2 = k, you take a square root of both sides and you are done. The trick is always the same: take half of the coefficient on x, square it, and add that amount to both sides. Half of b, squared, is exactly what turns x2 + bx into a perfect square trinomial, because (x + b/2)2 expands to x2 + bx + (b/2)2.
When the leading coefficient is not 1, factor it out of the x2 and x terms first. For 2x2 - 12x + 4, pull the 2 out of the first two terms to get 2(x2 - 6x) + 4, complete the square inside the parentheses using half of -6, then multiply that added constant by the 2 you factored out before adjusting the outside constant to keep the equation balanced.
Completing the square is not just a solving method. It produces vertex form, a(x - h)2 + k, which shows the vertex of the parabola directly. The quadratic formula is actually completing the square done once in general, on ax2 + bx + c = 0, so the two methods always agree.
Key terms
- Perfect square trinomial
- An expression like x2 + bx + (b/2)2 that factors exactly into (x + b/2)2.
- Vertex form
- The form a(x - h)2 + k, which shows a parabola's vertex at the point (h, k) directly.
- Completing the square
- Adding a constant to a quadratic expression so it becomes a perfect square binomial squared.
- Quadratic formula
- The formula x = (-b ± √(b2 - 4ac)) / (2a), derived by completing the square on ax2 + bx + c = 0.
Watch it explained
- How it worksHow do you complete the square when the leading coefficient is something other than 1?
- Worked exampleSolve x2+8x-9=0 by completing the square instead of factoring it.
- Worked exampleSolve 2x2-12x+4=0 by completing the square after dividing everything through by 2.
- Common mistakeWhy doesn't adding half of b squared to just one side keep the equation balanced?
- Going furtherWhat does completing the square hand you that the quadratic formula quietly hides?
Check yourself
End of topic.