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The Quadratic Formula

The Hook

The Sledgehammer.

What is it?

The universal formula to solve ANY quadratic equation (ax² + bx + c = 0).

Time Travel (History)

Derived to solve problems where factoring fails.

Cheat Code

Negative b, plus or minus root b squared minus 4ac, all over 2a.

Practice This Now!

The Field Guide

In Plain English: Factoring is nice, but it only works sometimes. This formula works ALWAYS. The part inside the square root (Discriminant) tells the future: if it's positive, 2 real answers. If negative, 2 imaginary answers.

In The Real World: Ballistics. Calculating exactly where a projectile will land, even with messy numbers.

How To Do It

  1. Make sure equation is = 0.
  2. Identify a, b, and c.
  3. Calculate Discriminant first: D = b² - 4ac.
  4. Plug into formula: x = (-b ± √D) / 2a.
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Booby Trap!

Sign errors. If b is negative (-5), then -b is positive (5). Students miss this constantly.

Real World Challenge
"A rocket path is -16t² + 100t. When does it hit the ground (height 0)?"
The Logic: -16t² + 100t = 0. Use formula. a=-16, b=100, c=0. t = 0 (start) and t = 6.25 seconds.

Training Drills

The Beast
x = -b ± √(b²-4ac) -------------- 2a
Sing it to 'Pop Goes the Weasel'.