The Quadratic Formula

Lesson 11 of 12 · 7 min · Free, no account needed

What you will learn

Completing the square with letters gives the quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a.
The discriminant b² − 4ac is what's left in the corner: positive means 2 roots, zero means 1, negative means none.
The formula works every time, even when the roots are not whole numbers, like 3 ± √2.

Where is x² − 6x + 7 equal to 0?
No graph this time. Drag the two pins to your best guesses.
Complete the square once, for every equation.
Press Step. The left column uses letters, the right one follows with b = −6 and c = 7.
Raise c. When is there exactly one root?
Turn c in x² − 6x + c. Watch the leftover corner b² − 4c and the roots.
The quadratic formula.
For any ax² + bx + c = 0, the quadratic formula gives the roots: x = (−b ± √(b² − 4ac)) ÷ 2a.
The part under the root, b² − 4ac, is the discriminant: what's left in the corner after completing the square.
Positive: two roots. Zero: one root. Negative: no roots, because no real number squares to a negative.
Build an equation the formula can't land.
Tap a slot, then a number. Make b² − 4ac negative.
How many roots? Sort by the discriminant.
Tap an equation, then its bin. Work out b² − 4ac in your head.
Game: kiss the ground.
a and b are given. Set c, then drop the curve. It should touch the ground at exactly one point.
Your equation, run through the formula in Python.
The program computes d = b² − 4ac and prints the roots only if d is not negative. Press Run.
Back to your first guess.
Your two pins for x² − 6x + 7 = 0. Tap Use the formula.

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