SOLUTION: There are two numbers with the following properties. 1) the second number is 3 more than the first number. 2) the product of the two numbers is 9 more than their sum. Which of t

Question 243543: There are two numbers with the following properties.
1) the second number is 3 more than the first number.
2) the product of the two numbers is 9 more than their sum.
Which of the following represents possible values of these two numbers?
A) -6,-3 B)-4,-1 C)-1,4 D)-3,6
*ALSO CAN YOU PLEASE SHOW ME THE STEPS TO SOLVE THE QUESTION? THANK YOU!
Answer by oberobic(2304) (Show Source): You can put this solution on YOUR website!
A key to simplifying word problems is to use as few unknowns as possible. Usually this means defining two (or more) variables in terms of one unknown.
.
For example, we are told there are two numbers. That suggests x & y. But we also are told that y is 3 more than x, which can be shown as x+3. So we can use this approach here.
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The product of the two numbers is 9 more than their sum.
x(x+3) = x + (x+3) + 9
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Multiplying the left-hand factor, we have
x^2 + 3x = x + (x+3) + 9
Collecting like terms:
x^2 + 3x = 2x + 12
Subtracting 2x from both sides:
x^2 + x = 12
Subtracting 12 from both sides:
x^2 + x - 12 = 0
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Now we have to decide if we can factor the quadratic. We see that 3*4 = 12 are separated by just 1. And since the 12 is negative, we use opposite signs: -3 and +4 = 1.
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(x-3)(x=4) = 0
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We have two candidate solutions: x=3 and x=-4.
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Testing x=3.
The consecutive terms would be 3 and 6.
The second number is now 3 more than the first
The product is 3*6 = 18.
The sum is 3+6=9, which is 9 less than 18.
So that checks.
.
Testing x = -4.
The terms would be -4 and -1.
The product = -4 * -1 = 4.
The sum = -4 + -1 = -5.
Are they 9 apart? Yes.
So that checks, too.
.
That means the values are:
-4, -1
and
3, 6
.
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