The complete guide — vertex form, solving equations, circles, and how to derive the quadratic formula from scratch.
Convert ax² + bx + c to a(x−h)² + k to find the vertex (h, k) — axis of symmetry and max/min value.
Solve any quadratic that won't factor. A backup method when factoring fails — always works.
Rewrite x² + y² + Dx + Ey + F = 0 into standard form (x−h)² + (y−k)² = r² to find center and radius.
A perfect square trinomial factors as a binomial squared:
x² + bx + (b/2)² = (x + b/2)²
x² − bx + (b/2)² = (x − b/2)²
Pattern Examples
x² + 6x + 9 = (x + 3)² [b=6, (b/2)²=9]
x² − 10x + 25 = (x − 5)² [b=10, (b/2)²=25]
x² + 3x + 9/4 = (x + 3/2)² [b=3, (b/2)²=9/4]
Rule: Take half the coefficient of x, square it — that's the term to add.
Make sure coefficient of x² is 1 (if not, divide everything by a)
Move constant (c) to the right side
Add (b/2)² to BOTH sides
Factor left side as (x + b/2)²
Take square root of both sides (include ± on right)
Solve for x
Worked Example: Solve x² + 8x − 9 = 0
Step 2: x² + 8x = 9
Step 3: (b/2)² = (8/2)² = 16 → x² + 8x + 16 = 9 + 16
Step 4: (x + 4)² = 25
Step 5: x + 4 = ±5
Step 6: x = −4 + 5 = 1 or x = −4 − 5 = −9
Same steps, but first divide the entire equation by a (or factor a out of the x terms).
Worked Example: Solve 2x² − 12x + 4 = 0
Divide by 2: x² − 6x + 2 = 0
Move constant: x² − 6x = −2
(b/2)² = (−6/2)² = 9 → x² − 6x + 9 = −2 + 9
(x − 3)² = 7
x − 3 = ±√7
x = 3 ± √7
Vertex Form Example: Write y = 3x² − 18x + 7 in vertex form
Factor 3 from x terms: y = 3(x² − 6x) + 7
Complete inside: (b/2)² = 9 → add and subtract inside:
y = 3(x² − 6x + 9 − 9) + 7
y = 3(x − 3)² − 3(9) + 7 = 3(x − 3)² − 27 + 7
y = 3(x − 3)² − 20 → Vertex: (3, −20)
Complete the square in both x and y to convert a general circle equation to standard form.
Example: x² + y² − 6x + 4y − 3 = 0
Group x and y: (x² − 6x) + (y² + 4y) = 3
Complete x: add (−6/2)² = 9 → (x² − 6x + 9)
Complete y: add (4/2)² = 4 → (y² + 4y + 4)
(x² − 6x + 9) + (y² + 4y + 4) = 3 + 9 + 4
(x − 3)² + (y + 2)² = 16
Center: (3, −2), Radius: √16 = 4
The quadratic formula comes from completing the square on the general form ax² + bx + c = 0:
ax² + bx + c = 0
x² + (b/a)x + c/a = 0 [÷ by a]
x² + (b/a)x = −c/a
x² + (b/a)x + (b/2a)² = −c/a + (b/2a)²
(x + b/2a)² = (b² − 4ac) / (4a²)
x + b/2a = ±√(b² − 4ac) / (2a)
x = (−b ± √(b² − 4ac)) / (2a) ✓
For ax² + bx + c: (1) Divide by a if a ≠ 1; (2) Move constant to the right side; (3) Add (b/2)² to both sides; (4) Factor the left side as a perfect square trinomial (x + b/2)²; (5) Solve for x by taking the square root of both sides. Example: x² + 6x + 5 = 0 → x² + 6x = −5 → x² + 6x + 9 = −5 + 9 → (x+3)² = 4 → x = −3 ± 2 → x = −1 or x = −5.
Divide the entire equation by a first, then proceed with the standard steps. Example: 2x² + 8x − 3 = 0 → x² + 4x − 3/2 = 0 → x² + 4x = 3/2 → x² + 4x + 4 = 3/2 + 4 → (x+2)² = 11/2 → x = −2 ± √(11/2). Alternatively, factor a out of only the x² and x terms before completing the square.
Completing the square has three main uses: (1) Converting a quadratic to vertex form y = a(x−h)² + k to find the vertex; (2) Solving quadratic equations that don't factor; (3) Writing circle equations in standard form (x−h)² + (y−k)² = r² to find center and radius. It's also how the quadratic formula is derived.
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