Perfect Square Trinomial
ax²+bx+c is a perfect square when b²=4ac. Then (√a·x+√c)² or (√a·x−√c)². (a+b)²=a²+2ab+b². (a−b)²=a²−2ab+b². Complete the square: add (b/2a)² to get perfect square.
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b²=4ac is the discriminant condition for a perfect square trinomial. Completing the square: add (b/2a)² to both sides of ax²+bx+c=0. x²+6x+9 = (x+3)² — middle term 6 = 2·x·3, last term 9 = 3².
Ready to run the numbers?
Why: Perfect square trinomials are key for completing the square, solving quadratics, and graphing parabolas. The discriminant b²−4ac=0 identifies them. Used in optimization and calculus.
How: Check: b²=4ac? If yes, factor as (√a·x±√c)². Generate: expand (px+q)². Complete: add (b/2a)²−c to ax²+bx+c, then factor.
Run the calculator when you are ready.
📐 Examples — Click to Load
Coefficients: ax² + bx + c
Coefficients & Values
Perfect Square Status
📐 Calculation Steps
For educational and informational purposes only. Verify with a qualified professional.
🧮 Fascinating Math Facts
b²=4ac means the quadratic has a double root — perfect square.
— Discriminant
x²+6x+9 = (x+3)². Middle coefficient 6 = 2·1·3.
— Example
📋 Key Takeaways
- • A perfect square trinomial has the form a²+2ab+b² = (a+b)² or a²-2ab+b² = (a-b)²
- • For ax²+bx+c to be a perfect square: b² = 4ac (discriminant = 0)
- • The middle term b must equal ±2√(ac) — twice the product of square roots of first and last terms
- • Completing the square adds (b/2)² to create a perfect square trinomial
- • Geometric interpretation: (a+b)² represents the area of a square with side length a+b
💡 Did You Know?
📖 The Pattern a²+2ab+b²
Expand (a+b)² = (a+b)(a+b) = a² + ab + ab + b² = a² + 2ab + b². So any trinomial matching this pattern factors as (a+b)². For (a-b)² we get a² - 2ab + b².
Example: x²+6x+9
a=x, b=3. Check: a²=x² ✓, 2ab=6x ✓, b²=9 ✓. So x²+6x+9 = (x+3)².
Connection to Completing the Square
To complete x²+8x: add (8/2)² = 16. Then x²+8x+16 = (x+4)². The constant is always (b/2)².
📐 Geometric Interpretation
A square with side (a+b) has area (a+b)². Split it into four parts: a square of side a (area a²), a square of side b (area b²), and two rectangles of area ab each. Total: a² + 2ab + b². This visual proof explains why the pattern works.
⚠️ Common Mistakes to Avoid
- Forgetting that b can be negative: (x-3)² = x² - 6x + 9, so b = -6.
- When a≠1, check b = ±2√(ac). For 4x²-12x+9: √4=2, √9=3, 2·2·3=12 ✓ (with minus).
- Not all trinomials with perfect square first and last terms are perfect squares — the middle term must match.
- Using b²=4ac: if a=1, c=9, then 4ac=36, so b²=36 means b=±6. Only ±6 works, not 5 or 7.
🔢 Quick Reference: Perfect Square Forms
| Form | Formula |
|---|---|
| Sum squared | (a+b)² = a² + 2ab + b² |
| Difference squared | (a-b)² = a² - 2ab + b² |
| Check condition | b² - 4ac = 0 |
| Complete the square | Add (b/2)² to c |
📊 Reference: Discriminant Check
| Condition | Meaning |
|---|---|
| b² - 4ac = 0 | Perfect square trinomial, one repeated root |
| b² - 4ac > 0 | Not perfect square, two distinct roots |
| b² - 4ac < 0 | Not perfect square, no real roots |
❓ FAQ
What is a perfect square trinomial?
A trinomial ax²+bx+c that can be written as (px+q)². It satisfies b²=4ac (discriminant = 0).
How do I check if a trinomial is a perfect square?
Calculate b²-4ac. If it equals 0, the trinomial is a perfect square. Also verify that √a and √c exist and b = ±2√(ac).
What do I add to complete x²+8x to a perfect square?
Add (8/2)² = 16. So x²+8x+16 = (x+4)². The rule: add (b/2)².
Why does (a+b)² = a²+2ab+b²?
Expand (a+b)(a+b) = a²+ab+ab+b². The two ab terms combine to 2ab.
Can 4x²-12x+9 be a perfect square?
Yes! √4=2, √9=3, and 2(2)(3)=12. With minus: (2x-3)² = 4x²-12x+9.
📝 Worked Examples
🎯 Factoring Shortcut
For ax²+bx+c where a and c are perfect squares: Let √a = p and √c = q. If b = 2pq (positive) then the form is (px+q)². If b = -2pq (negative) then the form is (px-q)². Example: 9x²+24x+16 → √9=3, √16=4, 2·3·4=24 ✓, so (3x+4)².
📌 Summary
A perfect square trinomial has the form (a+b)² = a²+2ab+b² or (a-b)² = a²-2ab+b². The key check is b² = 4ac (discriminant zero). Use this calculator to verify trinomials, expand binomials squared, or find what constant to add when completing the square. The geometric interpretation — area of a square — makes the pattern intuitive.
🎓 Practice Problems (Try in Calculator)
🔗 Next Steps
After mastering perfect square trinomials, explore the Completing the Square Calculator to convert quadratics to vertex form, or the Factoring Trinomials Calculator for general trinomial factoring.
⚠️ Disclaimer: This calculator checks, generates, and completes perfect square trinomials. For check mode, enter coefficients a, b, c of ax²+bx+c. For generate mode, enter p and q of (px+q)². For complete mode, enter a, b, and current c to find what to add.
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