To solve the quadratic equation [tex]\( x^2 + 8x - 33 = 0 \)[/tex], we follow a systematic approach applying the quadratic formula. The quadratic formula for solving the equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Given the coefficients:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 8 \)[/tex]
- [tex]\( c = -33 \)[/tex]
1. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ \Delta = 8^2 - 4 \cdot 1 \cdot (-33) \][/tex]
[tex]\[ \Delta = 64 + 132 \][/tex]
[tex]\[ \Delta = 196 \][/tex]
Since the discriminant is positive ([tex]\(\Delta > 0\)[/tex]), this indicates that there are two distinct real roots.
2. Calculate the roots:
The roots of the quadratic equation can be determined using:
[tex]\[ x_{1,2} = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substituting [tex]\(\Delta = 196\)[/tex] and the given values:
[tex]\[ x_{1,2} = \frac{-8 \pm \sqrt{196}}{2 \cdot 1} \][/tex]
[tex]\[ x_{1,2} = \frac{-8 \pm 14}{2} \][/tex]
Now compute the two possible values:
a. For the positive root ([tex]\( + \)[/tex]):
[tex]\[ x_1 = \frac{-8 + 14}{2} \][/tex]
[tex]\[ x_1 = \frac{6}{2} \][/tex]
[tex]\[ x_1 = 3 \][/tex]
b. For the negative root ([tex]\( - \)[/tex]):
[tex]\[ x_2 = \frac{-8 - 14}{2} \][/tex]
[tex]\[ x_2 = \frac{-22}{2} \][/tex]
[tex]\[ x_2 = -11 \][/tex]
Hence, the roots of the quadratic equation [tex]\( x^2 + 8x - 33 = 0 \)[/tex] are [tex]\( x_1 = 3 \)[/tex] and [tex]\( x_2 = -11 \)[/tex].
Additionally, the discriminant is [tex]\( 196 \)[/tex], confirming that the equation has two real and distinct roots.
So, the detailed solution is:
- Discriminant ([tex]\(\Delta\)[/tex]): 196
- Roots: [tex]\( x_1 = 3 \)[/tex], [tex]\( x_2 = -11 \)[/tex]
