Let

x

and

y

be the desired real numbers. Using the given information, we can write two equations, one for the sum of the numbers and one for the product of the numbers.

Sum : Product : x + y = - 15 x y = - 54

We can solve this system of equations using the Substitution Method. First, let's isolate

y

in the first equation and then substitute it into the second equation.

{x + y = - 15 x y = - 54 (I) (II)

SubEqn

$(I):$ $LHS - x = RHS - x$

{y = - 15 - x x y = - 54

Substitute

$(II):$ $y = - 15 - x$

{y = - 15 - x x (- 15 - x) = - 54

Distr

$(II):$ Distribute $x$

{y = - 15 - x - 15 x - x^{2} = - 54

AddEqn

$(II):$ $LHS + 54 = RHS + 54$

{y = - 15 - x - 15 x - x^{2} + 54 = 0

CommutativePropAdd

$(II):$ Commutative Property of Addition

{y = - 15 - x - x^{2} - 15 x + 54 = 0

Our next step is to solve the obtained quadratic equation.

- x^{2} - 15 x + 54 = 0 ⇕ - 1 x^{2} + (- 15) x + 54 = 0

We can do this by graphing. Let's begin by finding the axis of symmetry of the parabola using the formula

x = - \frac{b}{2 a} .

For our equation, we have that

a

=

- 1

and

b

=

- 15 .

x = - \frac{b}{2 a}

SubstituteII

$a = - 1$ , $b = - 15$

x = - \frac{- 1 5}{2 ( - 1 )}

▼

Simplify

MultPosNeg

$a (- b) = - a \cdot b$

x = - \frac{- 1 5}{- 2}

DivNegNeg

$\frac{- a}{- b} = \frac{a}{b}$

x = - \frac{1 5}{2}

CalcQuot

Calculate quotient

x = - 7.5

Next, we will make a table of values using

x

values around the axis of symmetry

x = - 7.5 .

$x$	$- x^{2} - 15 x + 54$	$y$
$- 18$	$- (- 18)^{2} - 15 (- 18) + 54$	$0$
$- 16$	$- (- 16)^{2} - 15 (- 16) + 54$	$38$
$- 7.5$	$- (- 7.5)^{2} - 15 (- 7.5) + 54$	$110.25$
$1$	$- (1)^{2} - 15 (1) + 54$	$38$
$3$	$- (3)^{2} - 15 (3) + 54$	$0$

Now, we will plot and connect the obtained points.

The roots of the equation are

- 18

and

3 .

Let's investigate if they satisfy the original problem.

Sum : Product : - 18 + 3 = - 15 ✓ - 18 (3) = - 54 ✓

Since the numbers satisfy both requirements stated in the exercise we have found the numbers we are looking for, and they are

- 18

and

3 .

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