# The product of two numbers is 144, and the sum of the two values is -7. What is the value of the smaller of the two numbers? A) 1 B) 5 C) -9 D) 14

Feb 8, 2017

The two numbers are 16 and -9, so the answer is C) -9

#### Explanation:

We have two unknown values - the greater integer and the smaller one. The two pieces of information must be used to create two equations that link the unknowns.

If you think of each equation as producing a line or curve on an $x , y$ plane, the answer(s) correspond to where they cross.

The equations are

$x \cdot y = 144$

and

$x + y = - 7$

The technique is to rewrite one equation so you can use it to substitute for one of the unknowns in the other equation. Here's how:

The second equation is written $y = x - 7$

Now, replace $y$ with $x - 7$ in the first equation:

#x(x-7) = 144

${x}^{2} - 7 x = 144$ becomes ${x}^{2} - 7 x - 144 = 0$

Use the quadratic formula (or factor it if you can):

$x = \frac{7 \pm \sqrt{{\left(- 7\right)}^{2} - 4 \left(1\right) \left(- 144\right)}}{2}$

$x = \frac{7 \pm 25}{2}$

which is true for $x = 16$ or $x = - 9$

Substitute into $x \cdot y = 144$ to get the value of $y$

$y = - 9$ or $y = 16$

Whichever way you look at it, the two numbers must be 16 and -9.