To solve the given equation [tex]\( x^2 + 2x = 15 \)[/tex] and determine the type of reasoning used, let's go through the steps and logic:
1. Rewrite the Equation:
Start with the given equation and move all terms to one side to get a standard quadratic form:
[tex]\[
x^2 + 2x - 15 = 0
\][/tex]
2. Identify the Coefficients:
In the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
a = 1, \quad b = 2, \quad c = -15
\][/tex]
3. Calculate the Discriminant:
The discriminant ([tex]\( \Delta \)[/tex]) of a quadratic equation is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values:
[tex]\[
\Delta = 2^2 - 4 \cdot 1 \cdot (-15) = 4 + 60 = 64
\][/tex]
4. Solve for the Roots:
The quadratic formula to find the roots [tex]\( x \)[/tex] of the equation is:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Plugging in the values:
[tex]\[
x = \frac{-2 \pm \sqrt{64}}{2 \cdot 1} = \frac{-2 \pm 8}{2}
\][/tex]
Calculate the two solutions:
[tex]\[
x_1 = \frac{-2 + 8}{2} = \frac{6}{2} = 3
\][/tex]
[tex]\[
x_2 = \frac{-2 - 8}{2} = \frac{-10}{2} = -5
\][/tex]
5. Check if [tex]\( x = 3 \)[/tex] is Indeed a Solution:
One of the solutions we found is [tex]\( x = 3 \)[/tex]. Thus, the statement [tex]\( x = 3 \)[/tex] is correct.
6. Determine the Type of Reasoning:
- Deductive Reasoning: This involves starting with a general statement and deriving a specific conclusion that logically follows from the general premises.
- In this problem, once we solve the equation exactly and find [tex]\( x = 3 \)[/tex], our conclusion follows directly and necessarily from the given premises without any assumptions or probabilities.
Given that the solution [tex]\( x = 3 \)[/tex] follows logically and necessarily from solving the quadratic equation, we conclude that the reasoning used is Deductive, valid.
