# Solvers

Solvers-virtuoso in many cases facilitate the work itself by resorting to simple algebraic transformations. For example, the calculation of 988² performed as follows: 988 x 988 = (988 + 12) x (988 – 12) + 12² = 1000 x 976 + 144 = 976144.

It is easy to see that solvers in this case doing the following algebraic manipulation:

a² = a² + b² – b² = (a + b)*(a - b) + b².

In practice, the solvers successfully used this formula for oral calculations. For example:

272 = (27 + 3)*(27 - 3) + 32 = 729

632 = 66*60 + 32 = 3969

For a quick squaring numbers ending in 5, the solvers uses the following method:

35² 3 х 4 = 12. (1225).

65² 6 х 7 = 42. (4225).

75² 7 х 8 = 56. (5625).

In this case, the number of tens must be multiplied by the number more per unit. Then, to the resulting product must be attributed the number 25.

The rule based on the following. If the number of tens of "a", then number can be represented as follows:

10а + 5.

The square of this number is:

100а² + 100а + 25 = 100a(a + 1) + 25.

The expression a (a + 1) is the product the number of tens and the next higher number. Multiply the number by 100 and added 25 - all the same that "attributed" to the number 25.

From the same reception it follows a simple way to build a number of square, consisting of whole and half. For example:

(3 1/2)² = 3,5² = 12,25 = 12 1/4.