SOLUTION: I really need help with this problem. The problem says to find two consecutive intergers whose product is 90, it must be solved algebraically. Thank You.

Question 115555: I really need help with this problem. The problem says to find two consecutive intergers whose product is 90, it must be solved algebraically. Thank You.
Found 2 solutions by jim_thompson5910, bucky:
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Remember, consecutive integers follow the pattern , , etc.

So if their product is 90, then

Distribute

Move all of the terms to the left side

Factor the left side (note: if you need help with factoring, check out this solver)

Now set each factor equal to zero:
or

or Now solve for x in each case

So our answer is
or

So our two numbers are either 9,10 or -10,-9
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
Two unknown consecutive integers can be represented by x and x+1 since the second integer must
be one greater than the first one ... because they are consecutive.
.
The product of these two integers is then given by:
.
x*(x + 1)
.
and this product must equal 90. This makes the equation become:
.
x*(x+1) = 90
.
And multiplying out the left side results in the equation:
.
x^2 + x = 90
.
Get rid of the 90 on the left side by subtracting 90 from both sides to get:
.
x^2 + x - 90 = 0
.
You can solve this by graphing it and seeing at what values of x the graph crosses the
x-axis. Or you can use the quadratic formula which is a short way of completing the square,
or you can factor it.
.
Let's use the quadratic formula which says that for a quadratic equation of the standard form:
.
ax^2 + bx + c = 0
.
the values of x that satisfy this equation are given by:
.

.
Compare our equation to the standard quadratic form. When you do you will see that "a" which
is the multiplier of x must be 1. b which is the multiplier of x must also be 1. And c which is
the constant must be -90. Substitute these values into the equation for x and you get:
.

.
The term inside the radical sign multiplies out to become . This changes the
answer to:
.

.
but the square root of 361 is 19. So you can replace the radical by 19 and the answers for x
are then:
.

.
Notice that the denominator multiplies out to 2 which makes the problem:
.

.
The two possible answers are then:
.
and
.

.
If x is -10, then the next consecutive integer is -10 + 1 = -9. This means the consecutive
integers -10 and -9 are one answer.
.
If x is +9, then the next consecutive integer is +9 + 1 = +10. This means the consecutive
integers +9 and +10 are another answer.
.
Check: -10 times -9 = +90. That checks. And +9 times +10 = +90. That also checks. So
our two answers are correct.
.
Hope this helps you with this problem.
.
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