Certainly! Let's break down the problem step by step:
We need to find two numbers that satisfy two conditions:
1. Their product is -32.
2. Their sum is 4.
Let's designate the two unknown numbers as [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
### Step 1: Set Up Equations
We can translate the conditions into the following equations:
1. [tex]\( x \times y = -32 \)[/tex]
2. [tex]\( x + y = 4 \)[/tex]
### Step 2: Express One Variable in Terms of the Other
From the second equation, we can express [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex]:
[tex]\[ y = 4 - x \][/tex]
### Step 3: Substitute into the First Equation
Substitute [tex]\( y = 4 - x \)[/tex] into the first equation:
[tex]\[ x \times (4 - x) = -32 \][/tex]
### Step 4: Simplify
Simplify the equation:
[tex]\[ 4x - x^2 = -32 \][/tex]
### Step 5: Rearrange into a Standard Quadratic Equation
Rearrange the equation to form a standard quadratic equation:
[tex]\[ x^2 - 4x - 32 = 0 \][/tex]
### Step 6: Solve the Quadratic Equation
To solve this quadratic equation, we factorize it. We look for two numbers that multiply to -32 and add to 4.
After checking the possible pairs, we find that:
[tex]\[ \begin{cases}
(x - 8)(x + 4) = 0
\end{cases} \][/tex]
### Step 7: Find the Roots of the Equation
Set each factor equal to zero and solve for [tex]\(x\)[/tex]:
[tex]\[ x - 8 = 0 \quad \Rightarrow \quad x = 8 \][/tex]
[tex]\[ x + 4 = 0 \quad \Rightarrow \quad x = -4 \][/tex]
### Step 8: Determine Corresponding [tex]\(y\)[/tex] Values
Using these [tex]\(x\)[/tex] values, we can find the corresponding [tex]\(y\)[/tex] values:
- If [tex]\( x = 8 \)[/tex], then [tex]\( y = 4 - 8 = -4 \)[/tex]
- If [tex]\( x = -4 \)[/tex], then [tex]\( y = 4 - (-4) = 8 \)[/tex]
### Final Answer
We have two pairs of numbers that satisfy both conditions:
[tex]\[ (x, y) = (8, -4) \][/tex]
[tex]\[ (x, y) = (-4, 8) \][/tex]
Thus, the two numbers you are looking for are: [tex]\( 8 \)[/tex] and [tex]\( -4 \)[/tex], or [tex]\( -4 \)[/tex] and [tex]\( 8 \)[/tex].
