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Solving Quadratic Equations
Rung 1 of 4 · Discover

Where Solving X² Comes From

A squared number isn't a mystery — it's the area of a square. So solving x² = k is just asking: "what's the side of a square with this area?" The answer is the square root.

NESA MA4-EQU-C-01Foundation concept

PlayDrag the corner to change the square's area, then read off the side. Area first, side second.
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The little ² just means "times itself". So is x × x — and that's exactly the area of a square whose side is x. A square 5 across holds 5 × 5 = 25 squares inside. That's the whole picture behind a quadratic.

Squaring, and Going Backwards

If someone hands you the side, finding the area is easy: square it. But a quadratic asks the reverse question — "I know the area, what was the side?" To go that way you need the operation that undoes squaring, and that's the square root, written . Set the square's area to 36 in the toy and the side comes out 6, because 6 × 6 = 36. So x² = 36 means x = √36 = 6.

Say it plainly: is the area of a square with side x. To solve x² = k you go backwards from area to side — and going backwards from a square is taking the square root: x = √k.

Why Square root, and Why √36 = 6

"Root" is just the old word for the side that the square grew from. Asking √36 is asking "what number, times itself, makes 36?" — and that's 6. You're not learning a brand-new trick; you're reversing the squaring you already know, the same way subtracting reverses adding.

Where This Is Heading

Linear equations were a balance scale — undo with inverse operations. A quadratic is the same idea with one new inverse: squaring is undone by square-rooting. Get that x² = k → x = √k picture solid here, and the method on the next rung is barely a step further.

Us, Thinking Out Loud

Why is finding a side from an area the "backwards" of squaring?

Where else have we gone backwards by using the opposite operation?