To solve the problem of finding two integers that add up to 2 and multiply to make -15, let's denote these integers as [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
We need to find two numbers that satisfy the following conditions:
1. [tex]\( x + y = 2 \)[/tex]
2. [tex]\( x \times y = -15 \)[/tex]
### Step 1: Write down the equations
We have the following system of equations based on the given conditions:
[tex]\[ x + y = 2 \][/tex]
[tex]\[ x \times y = -15 \][/tex]
### Step 2: Express one variable in terms of the other
From the first equation, we can express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[ y = 2 - x \][/tex]
### Step 3: Substitute into the second equation
Next, we substitute [tex]\( y \)[/tex] from the first equation into the second equation:
[tex]\[ x \times (2 - x) = -15 \][/tex]
### Step 4: Simplify the equation
Distribute [tex]\( x \)[/tex] in the equation:
[tex]\[ 2x - x^2 = -15 \][/tex]
### Step 5: Rearrange the equation to a standard quadratic form
Rearrange the equation to get:
[tex]\[ x^2 - 2x - 15 = 0 \][/tex]
### Step 6: Solve the quadratic equation
To solve for [tex]\( x \)[/tex], we can factor the quadratic equation. We look for two numbers that multiply to -15 and add up to -2.
The factors of -15 that can give a sum of -2 are -3 and 5:
[tex]\[ (x - 5)(x + 3) = 0 \][/tex]
### Step 7: Find the values of [tex]\( x \)[/tex]
Set each factor equal to zero:
[tex]\[ x - 5 = 0 \quad \text{or} \quad x + 3 = 0 \][/tex]
Solving these, we get:
[tex]\[ x = 5 \quad \text{or} \quad x = -3 \][/tex]
### Step 8: Find the corresponding [tex]\( y \)[/tex] values
Using [tex]\( y = 2 - x \)[/tex], we can find the corresponding [tex]\( y \)[/tex] values:
1. If [tex]\( x = 5 \)[/tex], then [tex]\( y = 2 - 5 = -3 \)[/tex]
2. If [tex]\( x = -3 \)[/tex], then [tex]\( y = 2 - (-3) = 5 \)[/tex]
### Step 9: Verify the solutions
We can check that both pairs (5, -3) and (-3, 5) satisfy the original conditions:
- For (5, -3):
- [tex]\( 5 + (-3) = 2 \)[/tex]
- [tex]\( 5 \times (-3) = -15 \)[/tex]
- For (-3, 5):
- [tex]\( -3 + 5 = 2 \)[/tex]
- [tex]\( -3 \times 5 = -15 \)[/tex]
Thus, the two integers that satisfy the conditions are:
[tex]\[ (5, -3) \quad \text{and} \quad (-3, 5) \][/tex]
