Sure, let's go through the process of finding the roots of the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] step-by-step and represent the process with a flowchart. We will also use the given coefficients [tex]\(a = 1\)[/tex], [tex]\(b = -3\)[/tex], and [tex]\(c = 2\)[/tex] to compute the roots.
### Flowchart
Start
1. Input: Read values for [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex].
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -3\)[/tex]
- [tex]\(c = 2\)[/tex]
2. Compute the Discriminant: [tex]\(D = b^2 - 4ac\)[/tex].
3. Check Discriminant:
- If [tex]\(D \geq 0\)[/tex], proceed to calculate the roots.
- If [tex]\(D < 0\)[/tex], output "No real roots" and End.
4. Calculate Roots:
- First root: [tex]\(x_1 = \frac{-b + \sqrt{D}}{2a}\)[/tex]
- Second root: [tex]\(x_2 = \frac{-b - \sqrt{D}}{2a}\)[/tex]
5. Output Roots: Display the roots [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex].
End
### Step-by-Step Solution
1. Read Coefficients
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -3\)[/tex]
- [tex]\(c = 2\)[/tex]
2. Calculate the Discriminant
[tex]\[
D = b^2 - 4ac
\][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
D = (-3)^2 - 4 \cdot 1 \cdot 2
\][/tex]
[tex]\[
D = 9 - 8
\][/tex]
[tex]\[
D = 1
\][/tex]
3. Check the Discriminant
- Since [tex]\(D = 1 \geq 0\)[/tex], there are real roots.
4. Calculate the Roots
- First root [tex]\(x_1\)[/tex]:
[tex]\[
x_1 = \frac{-b + \sqrt{D}}{2a}
\][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(D\)[/tex]:
[tex]\[
x_1 = \frac{-(-3) + \sqrt{1}}{2 \cdot 1}
\][/tex]
[tex]\[
x_1 = \frac{3 + 1}{2}
\][/tex]
[tex]\[
x_1 = \frac{4}{2}
\][/tex]
[tex]\[
x_1 = 2
\][/tex]
- Second root [tex]\(x_2\)[/tex]:
[tex]\[
x_2 = \frac{-b - \sqrt{D}}{2a}
\][/tex]
Substitute the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(D\)[/tex]:
[tex]\[
x_2 = \frac{-(-3) - \sqrt{1}}{2 \cdot 1}
\][/tex]
[tex]\[
x_2 = \frac{3 - 1}{2}
\][/tex]
[tex]\[
x_2 = \frac{2}{2}
\][/tex]
[tex]\[
x_2 = 1
\][/tex]
5. Output Roots
- The roots of the quadratic equation are [tex]\(x_1 = 2.0\)[/tex] and [tex]\(x_2 = 1.0\)[/tex].
### Final Answer
The roots of the equation [tex]\(ax^2 + bx + c = 0\)[/tex] with the given coefficients [tex]\(a = 1\)[/tex], [tex]\(b = -3\)[/tex], and [tex]\(c = 2\)[/tex] are [tex]\(2.0\)[/tex] and [tex]\(1.0\)[/tex].
