To solve the quadratic equation [tex]\( x^2 - 5x - 36 = 0 \)[/tex], follow these steps:
### Step 1: Identify the coefficients
For the quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex], we identify the coefficients:
- [tex]\( a = 1 \)[/tex] (coefficient of [tex]\( x^2 \)[/tex])
- [tex]\( b = -5 \)[/tex] (coefficient of [tex]\( x \)[/tex])
- [tex]\( c = -36 \)[/tex] (constant term)
### Step 2: Calculate the discriminant
The discriminant of a quadratic equation is given by the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values:
[tex]\[ \Delta = (-5)^2 - 4 \cdot 1 \cdot (-36) \][/tex]
[tex]\[ \Delta = 25 + 144 \][/tex]
[tex]\[ \Delta = 169 \][/tex]
### Step 3: Apply the quadratic formula
The quadratic formula to find the roots of the equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Using the calculated discriminant ([tex]\( \Delta = 169 \)[/tex]) and the coefficients:
[tex]\[ x = \frac{-(-5) \pm \sqrt{169}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{5 \pm 13}{2} \][/tex]
### Step 4: Calculate the roots
Now, we calculate the two possible roots:
#### Root 1:
[tex]\[ x_1 = \frac{5 + 13}{2} \][/tex]
[tex]\[ x_1 = \frac{18}{2} \][/tex]
[tex]\[ x_1 = 9 \][/tex]
#### Root 2:
[tex]\[ x_2 = \frac{5 - 13}{2} \][/tex]
[tex]\[ x_2 = \tripfrac{-8}{2} \][/tex]
[tex]\[ x_2 = -4 \][/tex]
### Conclusion
The solutions to the quadratic equation [tex]\( x^2 - 5x - 36 = 0 \)[/tex] are:
[tex]\[ x = 9 \text{ and } x = -4 \][/tex]
Thus, the roots of the equation are [tex]\( x = 9.0 \)[/tex] and [tex]\( x = -4.0 \)[/tex].
