Factoring with tiles
Slide b and c. You get one big x² tile, b strips (each is 1 by x) and c small unit tiles. The demo tries to pack them into one rectangle. When it works, read the two side lengths off the edges: they are the factors of x² + bx + c. When it cannot be done with whole numbers, it says so.
Mission. Set b = 7. Before you slide c, predict which values of c from 1 to 12 make a rectangle. Then test them all.
Answer
Only 6, 10 and 12. The two numbers must add to 7, so they are 1 and 6, 2 and 5, or 3 and 4. Their products are 6, 10 and 12. Every other c leaves a gap or spare tiles.
What's going on
A rectangle with sides x + p and x + q has four parts: an x² square, p strips down one side, q strips along the other, and a corner block of p × q unit tiles. So you need p + q strips and p × q unit tiles. Factoring x² + bx + c is the same hunt run backwards: find two numbers that add to b and multiply to c.
The list under the picture shows every pair of whole numbers that multiplies to c, with its sum. If one of the sums is b, that pair is highlighted and the rectangle closes up. If none is, no whole-number rectangle exists.
Tick Allow negatives and red tiles appear. A red tile counts as minus. A red strip cancels a blue strip, so 5 blue strips and 2 red strips are worth 3x. That is how x² + 3x − 10 can still be a rectangle, with sides x + 5 and x − 2.
Once you have the factors, solving the equation is quick. A product is zero only when one of its factors is zero, so (x + 3)(x + 4) = 0 means x = −3 or x = −4.
Sheets: Un-multiplying: the sum and product hunt and If a product is zero.