Understanding Completing the Square
Completing the square rewrites a quadratic ax² + bx + c as a perfect-square expression, a(x − h)² + k. This "vertex form" instantly shows the parabola's turning point (h, k) and provides a clean, mechanical way to solve for the roots without memorizing the quadratic formula separately — in fact, completing the square is exactly how the quadratic formula is derived.
The general formula
For any quadratic ax² + bx + c with a ≠ 0:
- Factor out a from the x² and x terms: a(x² + (b/a)x) + c.
- Add and subtract the completing term (b/2a)² inside the parentheses to build a perfect-square trinomial.
- Rewrite in vertex form: a(x − h)² + k, where h = −b/(2a) and k = c − b²/(4a).
- Discriminant: b² − 4ac determines the roots — positive gives two real roots, zero gives one repeated root, negative gives two complex roots.
Worked example with the default values
For x² + 6x + 5 (a = 1, b = 6, c = 5): the completing term is (6/2)² = 9. Adding and subtracting 9 gives x² + 6x + 9 − 9 + 5 = (x + 3)² − 4. So h = −3, k = −4, and the vertex form is (x + 3)² − 4. Since the discriminant is 6² − 4(1)(5) = 16 (positive), setting (x + 3)² − 4 = 0 gives (x + 3)² = 4, so x + 3 = ±2, meaning x = −1 or x = −5.
Solving for the roots
Once in vertex form, set a(x − h)² + k = 0 and isolate the square: (x − h)² = −k/a. If −k/a is zero or positive, take the square root of both sides and solve for x = h ± √(−k/a). This is algebraically identical to the quadratic formula x = (−b ± √(b² − 4ac)) / (2a) — completing the square is simply the derivation of that formula written out with your specific numbers.
Frequently Asked Questions
What is the formula for completing the square?
How do you find the roots by completing the square?
What does the discriminant (b² − 4ac) tell you?
Does completing the square still work if a is negative?
Common Tip Percentages
| Service | Typical Tip Range | Notes |
|---|---|---|
| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
| Food delivery | 10–20% | Consider extra for long distances |
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General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
|---|---|---|
| Needs | 50% | Rent, groceries, utilities, insurance |
| Wants | 30% | Dining out, entertainment, hobbies |
| Savings / Debt | 20% | Emergency fund, retirement, extra payments |
Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|---|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |