James Tanton’s Quadratics Solver

Step 1: First we have to ensure that the first term is a perfect square. If necessary, we can multiply the entire equation by some factor:

× 0× 0Skip!

If we multiply the entire equation by 0, we get

0x2+0x+0=0

We can already factorise 0x2 into a square.

Step 2: Next, we have to check that the second term is even, while also keeping the first term a square.

× 2× 4Skip!

0x2+0x+0=0

We can also fill in the remaining cells in the table.

Step 3: Finally, we have to ensure that the third term matches the value in the last square, by adding their difference to both sides of the equation:

+ 0+ 0Skip!

Step 4: The left-hand side of the equation is now a perfect square. We can read off the factorised version from the diagram:

1x+12=0

Step 5: Finally, we can take square roots of both sides.

1x+1=±0

Step 6: All that is left is to clean up the equation, and isolate x:

x=±0–1

There is only one solution: x=–1.