Certainly! Let's solve the given quadratic equation step by step. The standard form of a quadratic equation is typically written as:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
To find the solutions (roots) of this equation, we use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are coefficients from the quadratic equation.
Let's illustrate the process by assigning the following values to [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ a = 1 \][/tex]
[tex]\[ b = 1 \][/tex]
[tex]\[ c = 1 \][/tex]
### Step-by-Step Solution:
1. Identify the coefficients:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 1 \)[/tex]
- [tex]\( c = 1 \)[/tex]
2. 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] into the discriminant formula:
[tex]\[ \Delta = (1)^2 - 4(1)(1) \][/tex]
[tex]\[ \Delta = 1 - 4 \][/tex]
[tex]\[ \Delta = -3 \][/tex]
3. Analyze the discriminant:
- The discriminant [tex]\(\Delta = -3\)[/tex] is less than 0.
- This indicates that there are no real solutions to the quadratic equation, because the square root of a negative number is not real.
4. Conclusion:
Since the discriminant is negative, the quadratic equation [tex]\(x = \frac{-1 \pm \sqrt{-3}}{2(1)}\)[/tex] has no real solutions. Therefore, the equation has complex solutions.
In summary, for [tex]\(a = 1\)[/tex], [tex]\(b = 1\)[/tex], and [tex]\(c = 1\)[/tex], the discriminant is [tex]\(-3\)[/tex], which confirms that there are no real roots. Thus, the values of [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex] do not exist within the set of real numbers.
