To solve the quadratic equation [tex]\( m^2 - 5m - 14 = 0 \)[/tex], we want to find the values of [tex]\( m \)[/tex] that satisfy this equation. Here is the step-by-step solution process:
### Step 1: Identify a Quadratic Equation
The given quadratic equation is:
[tex]\[ m^2 - 5m - 14 = 0 \][/tex]
### Step 2: Factor the Quadratic Equation
We will factor this quadratic equation, which involves finding two numbers that multiply to give the constant term [tex]\(-14\)[/tex] and add up to give the coefficient of the linear term [tex]\(-5\)[/tex].
The factors of [tex]\(-14\)[/tex] that add up to [tex]\(-5\)[/tex] are [tex]\(-7\)[/tex] and [tex]\(2\)[/tex].
### Step 3: Write the Equation in Factored Form
Using the factors [tex]\(-7\)[/tex] and [tex]\(2\)[/tex], the quadratic equation can be written in its factored form as:
[tex]\[ (m - 7)(m + 2) = 0 \][/tex]
### Step 4: Solve for [tex]\( m \)[/tex]
The factored equation [tex]\( (m - 7)(m + 2) = 0 \)[/tex] gives us two potential solutions:
[tex]\[ m - 7 = 0 \quad \text{or} \quad m + 2 = 0 \][/tex]
Solving these equations for [tex]\( m \)[/tex], we get:
[tex]\[ m = 7 \][/tex]
[tex]\[ m = -2 \][/tex]
### Step 5: State the Solutions
Thus, the values of [tex]\( m \)[/tex] that satisfy the quadratic equation [tex]\( m^2 - 5m - 14 = 0 \)[/tex] are:
[tex]\[ m = 7 \][/tex]
[tex]\[ m = -2 \][/tex]
Hence, the correct answer is:
[tex]\[ x = 7 \ \text{and} \ x = -2 \][/tex]
The answer is:
[tex]\[
\boxed{x=7 \text{ and } x=-2}
\][/tex]
