Packets  /  Completing the Square
Grades 9-11Algebra

Completing the Square

Learn to complete the square: turn any quadratic into a perfect square to solve equations, find the vertex, and see where the quadratic formula comes from.

1Perfect Square Trinomials (Review)

Key idea: A perfect square trinomial is exactly what you get from squaring a binomial: x^2 + 2bx + b^2 = (x + b)^2. To spot one, check that the constant equals the square of half the middle coefficient.
  1. Recall the pattern: (x + b)2 = x2 + 2bx + b2.
  2. Take half of the x-coefficient: b = (middle coefficient) / 2.
  3. Square that half: the constant term must equal b2.
  4. If it matches, factor as (x + b)2, keeping the sign of the middle term.
Worked example
x2 + 10x + 25
Half of 10 is 5, and 52 = 25, which matches the constant. So x2 + 10x + 25 = (x + 5)2.
Worked example
x2 – 14x + 49
Half of -14 is -7, and (-7)2 = 49. The middle term is negative, so x2 – 14x + 49 = (x – 7)2. Check: (x – 7)2 = x2 – 14x + 49.
Watch out: x^2 + 10x + 24 is NOT a perfect square (24 is not 5^2) — never force the pattern. The constant must equal the square of half the middle coefficient.

2Completing the Square (a = 1)

Key idea: Add (b/2)^2 to both sides to turn x^2 + bx into the perfect square (x + b/2)^2.
  1. Move the constant term to the right side: x2 + bx = -c.
  2. Compute (b/2)2 — halve the x-coefficient, then square it.
  3. Add that number to BOTH sides of the equation.
  4. Write the left side as (x + b/2)2 and simplify the right side.
Worked example
Complete the square for x2 + 6x = 7
622 = 9. Add 9 to both sides: x2 + 6x + 9 = 7 + 9, so (x + 3)2 = 16.
Worked example
Complete the square for x2 – 8x + 12 = 0
Move the constant: x2 – 8x = -12. Then -822 = 16. Add 16 to both sides: (x – 4)2 = -12 + 16 = 4.
Watch out: The number you add must go on BOTH sides. Adding (b/2)^2 to only one side changes the equation and breaks every step after it.

3Completing the Square (a is not 1)

Key idea: If the x^2 coefficient is not 1, divide every term by it first — then the a = 1 steps work exactly as before.
  1. Divide every term on both sides by a (the x2 coefficient).
  2. Move the constant term to the right side.
  3. Add (b/2)2 to both sides, using the NEW b from after dividing.
  4. Write the left side as a perfect square and solve.
Worked example
Solve 2x2 + 12x + 10 = 0
Divide everything by 2: x2 + 6x + 5 = 0. Move the constant: x2 + 6x = -5. Add 622 = 9 to both sides: (x + 3)2 = 4. So x + 3 = ±2, giving x = -1 or x = -5.
Watch out: Divide BEFORE computing (b/2)^2. In 2x^2 + 12x, the b you halve is 6 (after dividing), not 12 — using (12/2)^2 = 36 here is wrong.

4Vertex Form of Quadratics

Key idea: Completing the square rewrites y = ax^2 + bx + c as y = a(x – h)^2 + k, where (h, k) is the vertex — the highest or lowest point of the parabola.
  1. Move the constant away from the x terms.
  2. Factor a out of the x terms if a is not 1.
  3. Complete the square inside, balancing any number you add.
  4. Collect constants: y = a(x – h)2 + k. The vertex is (h, k).
Worked example
Write y = x2 + 6x + 5 in vertex form
x2 + 6x needs 622 = 9. Add and subtract it: y = x2 + 6x + 9 – 9 + 5 = (x + 3)2 – 4. The vertex is (-3, -4).
Worked example
Find the vertex of y = x2 – 10x + 21
-1022 = 25, so y = (x – 5)2 – 25 + 21 = (x – 5)2 – 4. The vertex is (5, -4).
Watch out: In y = (x – h)^2 + k, h takes the OPPOSITE sign of what you see: (x + 3)^2 means h = -3, so the vertex is (-3, k), not (3, k).

5Solving Quadratics by Completing the Square

Key idea: Complete the square to get (x + b/2)^2 = a number, then take the square root of both sides — with ± — and solve. This method works even when factoring is impossible.
  1. Complete the square (divide by a first if it is not 1).
  2. Write the equation as (x + b/2)2 = N.
  3. Take the square root of both sides: x + b/2 = ±√N.
  4. Solve for x. If N is negative, there are no real solutions.
Worked example
Solve x2 – 6x + 7 = 0 (this one cannot be factored)
x2 – 6x = -7. Add -622 = 9 to both sides: (x – 3)2 = 2. Take square roots: x – 3 = ±√2, so x = 3 + √2 or x = 3 – √2.
Worked example
Solve x2 + 2x + 5 = 0
x2 + 2x = -5. Add 222 = 1 to both sides: (x + 1)2 = -4. A real square cannot be negative, so this equation has no real solutions.
Watch out: Forgetting the ± is the classic error: (x – 3)^2 = 2 gives TWO answers, x = 3 + √2 and x = 3 – √2.

6Connection to the Quadratic Formula

Key idea: The quadratic formula is not magic — it is exactly what you get by completing the square on the general equation ax^2 + bx + c = 0.
  1. Start with ax2 + bx + c = 0 and divide everything by a.
  2. Move c/a to the right side.
  3. Add (b/(2a))2 to both sides and write the left side as a perfect square.
  4. Take square roots (with ±) and solve for x — the result is the quadratic formula.
Worked example
Complete the square on the general equation ax2 + bx + c = 0
Divide by a: x2 + (b/a)x + c/a = 0. Move the constant: x2 + (b/a)x = -c/a. Add (b/(2a))2 to both sides: (x + b/(2a))2 = (b2 – 4ac)/(4a2). Taking square roots and solving for x gives x = (-b ± √(b2 – 4ac))/(2a) — the quadratic formula.
Watch out: The formula and completing the square are the same method in disguise — if you ever forget the formula on a test, just complete the square instead.
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60-Second Challenge

How many completing the square problems can you solve in 60 seconds?