Actually, double-check discriminant: 7² + 4×3×220 = 49 + 2640 = 2689, √2689 = ? - Treasure Valley Movers

Actual Double-Check: Solving Discriminant Expression 7² + 4×3×220 = 2689 – What’s √2689?

In mathematical problem-solving, accuracy matters at every step—especially when working with discriminants or complex expressions. Recently, a key verification step sparked attention: calculating the discriminant formed by the expression 7² + 4×3×220, followed by finding √2689. Let’s walk through this process carefully and validate what’s truly happening under the math.

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Step 1: Evaluate the Expression Inside the Discriminant

We start with:
7² + 4 × 3 × 220

Following the order of operations (PEMDAS/BODMAS):

1. Compute 7² = 49
   
2. Compute 4 × 3 = 12, then 12 × 220 = 2640
   
3. Add the results:
   49 + 2640 = 2689

So, the discriminant simplified expression is indeed 2689.

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Step 2: Calculate √2689 – What Value Does It Yield?

Now, we compute √2689, the square root of 2689. Unlike perfect squares (e.g., √256 = 16), 2689 is not a perfect square, so the square root is irrational.

Let’s estimate:

• 51² = 2601
  
• 52² = 2704

Since 2601 < 2689 < 2704, it follows that:
51 < √2689 < 52

Using linear approximation or a calculator for better precision:
√2689 ≈ 51.86 (rounded to two decimal places)

This decimal precision is useful in algebraic contexts—especially when checking discriminant properties in quadratic equations.

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Why This Verification Matters

In quadratic equations of the form ax² + bx + c = 0, the discriminant D = b² − 4ac determines the nature of roots:

• If D > 0: two distinct real roots
  
• If D = 0: one real root (repeated)
  
• If D < 0: complex roots

In our expression 7² + 4×3×220, this calculation leads to D = 2689, so D > 0 implies two distinct real roots—critical information for solving or analyzing quadratic models in science, engineering, or finance.

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Conclusion