Sure, let's solve the quadratic equation [tex]\( x^2 - 4x - 21 = 0 \)[/tex] step by step.
### Step 1: Identify coefficients
For the quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex], the coefficients are:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -4\)[/tex]
- [tex]\(c = -21\)[/tex]
### Step 2: Calculate the discriminant
The discriminant [tex]\( \Delta \)[/tex] of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Plugging in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = (-4)^2 - 4 \cdot 1 \cdot (-21)
\][/tex]
Simplify the calculation:
[tex]\[
\Delta = 16 + 84 = 100
\][/tex]
So, the discriminant is [tex]\( \Delta = 100 \)[/tex].
### Step 3: Determine the number of roots
- If [tex]\( \Delta > 0 \)[/tex], the quadratic equation has 2 distinct real roots.
- If [tex]\( \Delta = 0 \)[/tex], there is exactly 1 real root (a repeated root).
- If [tex]\( \Delta < 0 \)[/tex], there are no real roots (the roots are complex).
Since [tex]\( \Delta = 100 \)[/tex] which is greater than 0, the quadratic equation has 2 distinct real roots.
### Step 4: Solve the quadratic equation
The roots of the quadratic equation can be found using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substitute [tex]\(b = -4\)[/tex], [tex]\(a = 1\)[/tex], and [tex]\(\Delta = 100\)[/tex] into the formula:
[tex]\[
x = \frac{-(-4) \pm \sqrt{100}}{2 \cdot 1} = \frac{4 \pm 10}{2}
\][/tex]
### Step 5: Simplify to find the roots
There are two solutions based on the [tex]\(\pm\)[/tex] sign:
1. For the [tex]\(+\)[/tex] case:
[tex]\[
x_1 = \frac{4 + 10}{2} = \frac{14}{2} = 7.0
\][/tex]
2. For the [tex]\(-\)[/tex] case:
[tex]\[
x_2 = \frac{4 - 10}{2} = \frac{-6}{2} = -3.0
\][/tex]
### Final Result
The discriminant is [tex]\(100\)[/tex], indicating that there are 2 distinct real roots. The solutions to the quadratic equation [tex]\( x^2 - 4x - 21 = 0 \)[/tex] are:
[tex]\[
x_1 = 7.0 \quad \text{and} \quad x_2 = -3.0
\][/tex]
Thus, the result is:
[tex]\[
(100, 2, 7.0, -3.0)
\][/tex]
