Completing the Square
Completing the square is an algebraic technique for transforming a quadratic expression ax² + bx + c into the vertex form a(x − h)² + k by adding and subtracting a carefully chosen constant. This method reveals the vertex of a parabola directly and is used to derive the quadratic formula, solve quadratic equations, and convert conic section equations to standard form. It is a foundational technique in algebra, calculus (completing the square for integration), and analytic geometry.
Key Formula
ax² + bx + c = a(x + b/(2a))² − (b²−4ac)/(4a)
LaTeX: ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2 - 4ac}{4a}
| Symbol | Meaning | Unit |
|---|---|---|
| a | coefficient of x² | dimensionless |
| b | coefficient of x | dimensionless |
| c | constant term | dimensionless |
| h | x-coordinate of vertex: h = −b/(2a) | dimensionless |
| k | y-coordinate of vertex: k = c − b²/(4a) | dimensionless |
Worked Example
Problem
Solve 2x² − 8x + 3 = 0 by completing the square.
Solution
Step 1: Divide through by 2: x² − 4x + 3/2 = 0. Step 2: Move constant: x² − 4x = −3/2. Step 3: Add (−4/2)² = 4 to both sides: x² − 4x + 4 = −3/2 + 4 = 5/2. Step 4: Write as perfect square: (x − 2)² = 5/2. Step 5: Take square roots: x − 2 = ±√(5/2) = ±√10/2. Step 6: Solve: x = 2 ± √10/2.
Answer
x = 2 ± √10/2 ≈ 2 ± 1.581, so x ≈ 3.581 or x ≈ 0.419
Steps for Completing the Square (Monic Quadratic x² + bx + c)
| Step | Action | Expression After Step | Notes |
|---|---|---|---|
| 1 | Write quadratic | x² + bx + c | Coefficient of x² must be 1 |
| 2 | Move constant | x² + bx = −c | Isolate x terms |
| 3 | Add (b/2)² to both sides | x² + bx + (b/2)² = −c + (b/2)² | Creates perfect square |
| 4 | Factor left side | (x + b/2)² = −c + b²/4 | Perfect square trinomial |
| 5 | Solve for x | x + b/2 = ±√(−c + b²/4) | Take square root both sides |
| 6 | Isolate x | x = −b/2 ± √(b²/4 − c) | Final solutions |
Interactive Tools
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Related Terms
Quadratic Formula
The quadratic formula is an algebraic formula that gives the solutions (roots) of any quadratic equation ax² + bx + c = 0 directly in terms of its coefficients a, b, and c. It is derived by completing the square on the general quadratic and is the most reliable method for solving quadratics, working even when factoring over integers is impossible. The formula also reveals the nature of the roots through the discriminant b² − 4ac.
Discriminant
The discriminant of a quadratic equation ax² + bx + c = 0 is the expression b² − 4ac, denoted by the Greek letter Δ (delta) or D. The sign of the discriminant determines the nature and number of roots: positive means two distinct real roots, zero means one repeated real root, and negative means two complex (non-real) conjugate roots. The discriminant is the part under the square root sign in the quadratic formula, making it the key indicator of solvability over the real numbers.
The phrase "completing the square" is a literal geometric description: adding a smaller square region to an L-shaped figure to complete a larger square. The method was known to ancient Babylonian mathematicians (c. 2000 BCE) and was formalised algebraically by the Persian mathematician al-Khwarizmi in his 9th-century work "Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala."