To solve the quadratic equation [tex]\( m^2 - 5m - 14 = 0 \)[/tex] using the quadratic formula, we follow these steps:
### Step 1: Identify the coefficients
First, identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] from the quadratic equation [tex]\( am^2 + bm + c = 0 \)[/tex].
In this case:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -5 \)[/tex]
- [tex]\( c = -14 \)[/tex]
### Step 2: Calculate the discriminant
The discriminant [tex]\( \Delta \)[/tex] is given by the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Plugging in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ \Delta = (-5)^2 - 4 \cdot 1 \cdot (-14) = 25 + 56 = 81 \][/tex]
### Step 3: Use the quadratic formula
The quadratic formula is:
[tex]\[ m = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute the values of [tex]\( b \)[/tex], [tex]\( \Delta \)[/tex], and [tex]\( a \)[/tex]:
[tex]\[ m = \frac{-(-5) \pm \sqrt{81}}{2 \cdot 1} = \frac{5 \pm 9}{2} \][/tex]
### Step 4: Solve for the roots
Now, calculate the two possible values of [tex]\( m \)[/tex]:
- For the positive square root:
[tex]\[ m_1 = \frac{5 + 9}{2} = \frac{14}{2} = 7 \][/tex]
- For the negative square root:
[tex]\[ m_2 = \frac{5 - 9}{2} = \frac{-4}{2} = -2 \][/tex]
### Final Answer
The solutions to the quadratic equation [tex]\( m^2 - 5m - 14 = 0 \)[/tex] are:
[tex]\[ m_1 = 7 \][/tex]
[tex]\[ m_2 = -2 \][/tex]
In summary:
- The discriminant [tex]\( \Delta \)[/tex] is 81.
- The roots of the equation are [tex]\( m = 7 \)[/tex] and [tex]\( m = -2 \)[/tex].
