Answer :
Let's solve the given problem step-by-step.
### Part 1: Determine the nature of the roots of the quadratic equation [tex]\(2x^2 - 4x + 3 = 0\)[/tex]
To determine the nature of the roots of a quadratic equation of the form [tex]\(ax^2 + bx + c = 0\)[/tex], we can use the discriminant, which is calculated as [tex]\( \Delta = b^2 - 4ac \)[/tex].
For the equation [tex]\(2x^2 - 4x + 3 = 0\)[/tex]:
1. Identify the coefficients:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = -4\)[/tex]
- [tex]\(c = 3\)[/tex]
2. Calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values, we get:
[tex]\[ \Delta = (-4)^2 - 4 \cdot 2 \cdot 3 = 16 - 24 = -8 \][/tex]
3. Analyze the discriminant:
- If [tex]\(\Delta > 0\)[/tex], the equation has two distinct real roots.
- If [tex]\(\Delta = 0\)[/tex], the equation has two equal (repeated) real roots.
- If [tex]\(\Delta < 0\)[/tex], the equation has two complex roots.
Since [tex]\(\Delta = -8\)[/tex] which is less than zero, the quadratic equation [tex]\(2x^2 - 4x + 3 = 0\)[/tex] has complex roots.
### Part 2: Find the values of [tex]\(a\)[/tex] such that the quadratic equation [tex]\(9x^2 - 3ax + 1 = 0\)[/tex] has equal roots
For the quadratic equation [tex]\(9x^2 - 3ax + 1 = 0\)[/tex] to have equal roots, its discriminant must be zero. Let's find the discriminant for this equation.
1. Identify the coefficients:
- [tex]\(a = 9\)[/tex]
- [tex]\(b = -3a\)[/tex]
- [tex]\(c = 1\)[/tex]
2. Set the discriminant to zero:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the coefficient values, we get:
[tex]\[ \Delta = (-3a)^2 - 4 \cdot 9 \cdot 1 = 9a^2 - 36 \][/tex]
For the equation to have equal roots:
[tex]\[ 9a^2 - 36 = 0 \][/tex]
3. Solve for [tex]\(a\)[/tex]:
[tex]\[ 9a^2 = 36 \][/tex]
[tex]\[ a^2 = 4 \][/tex]
[tex]\[ a = \pm 2 \][/tex]
Therefore, the values of [tex]\(a\)[/tex] for which the quadratic equation [tex]\(9x^2 - 3ax + 1 = 0\)[/tex] has equal roots are [tex]\(\boxed{-2 \text{ and } 2}\)[/tex].
### Part 1: Determine the nature of the roots of the quadratic equation [tex]\(2x^2 - 4x + 3 = 0\)[/tex]
To determine the nature of the roots of a quadratic equation of the form [tex]\(ax^2 + bx + c = 0\)[/tex], we can use the discriminant, which is calculated as [tex]\( \Delta = b^2 - 4ac \)[/tex].
For the equation [tex]\(2x^2 - 4x + 3 = 0\)[/tex]:
1. Identify the coefficients:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = -4\)[/tex]
- [tex]\(c = 3\)[/tex]
2. Calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values, we get:
[tex]\[ \Delta = (-4)^2 - 4 \cdot 2 \cdot 3 = 16 - 24 = -8 \][/tex]
3. Analyze the discriminant:
- If [tex]\(\Delta > 0\)[/tex], the equation has two distinct real roots.
- If [tex]\(\Delta = 0\)[/tex], the equation has two equal (repeated) real roots.
- If [tex]\(\Delta < 0\)[/tex], the equation has two complex roots.
Since [tex]\(\Delta = -8\)[/tex] which is less than zero, the quadratic equation [tex]\(2x^2 - 4x + 3 = 0\)[/tex] has complex roots.
### Part 2: Find the values of [tex]\(a\)[/tex] such that the quadratic equation [tex]\(9x^2 - 3ax + 1 = 0\)[/tex] has equal roots
For the quadratic equation [tex]\(9x^2 - 3ax + 1 = 0\)[/tex] to have equal roots, its discriminant must be zero. Let's find the discriminant for this equation.
1. Identify the coefficients:
- [tex]\(a = 9\)[/tex]
- [tex]\(b = -3a\)[/tex]
- [tex]\(c = 1\)[/tex]
2. Set the discriminant to zero:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the coefficient values, we get:
[tex]\[ \Delta = (-3a)^2 - 4 \cdot 9 \cdot 1 = 9a^2 - 36 \][/tex]
For the equation to have equal roots:
[tex]\[ 9a^2 - 36 = 0 \][/tex]
3. Solve for [tex]\(a\)[/tex]:
[tex]\[ 9a^2 = 36 \][/tex]
[tex]\[ a^2 = 4 \][/tex]
[tex]\[ a = \pm 2 \][/tex]
Therefore, the values of [tex]\(a\)[/tex] for which the quadratic equation [tex]\(9x^2 - 3ax + 1 = 0\)[/tex] has equal roots are [tex]\(\boxed{-2 \text{ and } 2}\)[/tex].
