The product of two consecutive odd integers is 783. How do you find the integers?
3 Answers
Here's how you can do that.
Explanation:
The problem tells you that the product of two consecutive odd integers is equal to
Right from the start, you know that you can get from the smaller number to the bigger number by adding
You need to add
#"odd number" + 1 = "the consecutive even number"" "color(red)(xx)#
#"odd number" + 2 = "the consecutive odd number"" "color(darkgreen)(sqrt())#
So, if you take
#x + 2#
is the second number, which means that you have
#x * (x+2) = 783#
SIDE NOTE You can also go with
#(x-2) + 2 = x#
as the second number, the answer must come out the same.
This is equivalent to
#x^2 + 2x = 783#
Rearrange to quadratic equation form
#x^2 + 2x - 783 = 0#
Use the quadratic formula to find the two values of
#x_(1,2) = (-2 +- sqrt( 2^2 - 4 * 1 * (-783)))/(2 * 1)#
#x_(1,2) = (-2 +- sqrt(3136))/2#
#x_(1,2) = (-2 +- 56)/2 implies {( x_1 = (-2 - 56)/2 = -29), (x_2 = (-2 + 56)/2 = 27) :}#
Now, you have two valid solution sets here.
#"For"color(white)(.) x = -29#
# -29" "# and#" " - 29 + 2 = -27# Check:
#(-29) * (-27) = 783" "color(darkgreen)(sqrt())#
#"For"color(white)(.) x = 27#
# 27" "# and#" " 27 + 2 = 29# Check:
#27 * 29 = 783" "color(darkgreen)(sqrt())#
There are two solutions:
#27, 29#
and
#-29, -27#
Explanation:
One method goes as follows.
I will use the difference of squares identity:
#a^2-b^2 = (a-b)(a+b)#
Let
Then:
#783 = (n-1)(n+1) = n^2-1#
Subtract
#0 = n^2-784 = n^2-28^2 = (n-28)(n+28)#
So
There are therefore two possible pairs of consecutive odd integers:
#27, 29#
and:
#-29, -27#
Find
Explanation:
We know from the question that
We also know that the two factors are very close together because they are consecutive odd numbers.
If you consider factor pairs you will find that the closer factors are, the smaller is their sum or difference.
The factors which are furthest apart are
The factors which have the smallest sum or difference are the square roots. The square root of a number is the factor exactly in the middle if factors arranged in order.
The factors we are looking for must be very close to
Test odd numbers on either side of
Remember that the odd numbers can be negative as well.