To square 75, one can multiply (70)(80) and add 25 to that product. Can someone explain how this would work, without calculating?
Certainly! To understand why multiplying (70)(80) and adding 25 gives the square of 75, we can use a concept called the square of a binomial.
The square of a binomial can be expressed as:
(a + b)^2 = a^2 + 2ab + b^2
In this case, a = 70 and b = 5. Therefore, we can write the square of 75 as:
(70 + 5)^2 = 70^2 + 2(70)(5) + 5^2
Now, let's simplify this expression. First, we square 70 and 5:
70^2 = 4900
5^2 = 25
Next, we calculate the product of 2(70)(5):
2(70)(5) = 700
Finally, we substitute these values back into the original formula:
(70 + 5)^2 = 4900 + 700 + 25
Calculating the sum:
4900 + 700 + 25 = 5625
Therefore, the square of 75 is 5625.