Answer :
Let's solve the equation [tex]\(2 x^2 - 13 x + 8 = 0\)[/tex] to find its roots and then compare them with the given choices.
### Step-by-Step Solution:
1. Write the Quadratic Equation:
The equation we need to solve is:
[tex]\[ 2 x^2 - 13 x + 8 = 0 \][/tex]
2. Use the Quadratic Formula:
The quadratic formula for an equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\(a = 2\)[/tex], [tex]\(b = -13\)[/tex], and [tex]\(c = 8\)[/tex].
3. Calculate the Discriminant:
The discriminant [tex]\(\Delta\)[/tex] is calculated as:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values:
[tex]\[ \Delta = (-13)^2 - 4 \cdot 2 \cdot 8 = 169 - 64 = 105 \][/tex]
4. Calculate the Roots:
Using the quadratic formula:
[tex]\[ x = \frac{-(-13) \pm \sqrt{105}}{2 \cdot 2} \][/tex]
Simplifying further:
[tex]\[ x = \frac{13 \pm \sqrt{105}}{4} \][/tex]
5. Approximate the Roots:
Calculating the approximate values:
[tex]\[ \sqrt{105} \approx 10.246 \][/tex]
Therefore:
[tex]\[ x_1 = \frac{13 + 10.246}{4} \approx \frac{23.246}{4} \approx 5.811 \][/tex]
[tex]\[ x_2 = \frac{13 - 10.246}{4} \approx \frac{2.754}{4} \approx 0.689 \][/tex]
6. Compare with the Given Choices:
Now we compare the approximate roots with the choices:
[tex]\[ \text{Choice (a)}: x = -13.13, \, 3.25 \quad \text{(This does not match our values)} \][/tex]
[tex]\[ \text{Choice (b)}: x = 0.4, \, 0.67 \quad \text{(This does not match our values)} \][/tex]
[tex]\[ \text{Choice (c)}: x = 0.69, \, 5.81 \quad \text{(This closely matches our values: } 0.689 \approx 0.69 \text{ and } 5.811 \approx 5.81) \][/tex]
[tex]\[ \text{Choice (d)}: x = 1.38, \, 11.63 \quad \text{(This does not match our values)} \][/tex]
### Conclusion:
The approximate roots of the equation [tex]\(2 x^2 - 13 x + 8 = 0\)[/tex] are matched correctly by choice (c):
[tex]\[ x = 0.69 \text{ and } 5.81 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{\text{c}} \][/tex]
### Step-by-Step Solution:
1. Write the Quadratic Equation:
The equation we need to solve is:
[tex]\[ 2 x^2 - 13 x + 8 = 0 \][/tex]
2. Use the Quadratic Formula:
The quadratic formula for an equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, [tex]\(a = 2\)[/tex], [tex]\(b = -13\)[/tex], and [tex]\(c = 8\)[/tex].
3. Calculate the Discriminant:
The discriminant [tex]\(\Delta\)[/tex] is calculated as:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values:
[tex]\[ \Delta = (-13)^2 - 4 \cdot 2 \cdot 8 = 169 - 64 = 105 \][/tex]
4. Calculate the Roots:
Using the quadratic formula:
[tex]\[ x = \frac{-(-13) \pm \sqrt{105}}{2 \cdot 2} \][/tex]
Simplifying further:
[tex]\[ x = \frac{13 \pm \sqrt{105}}{4} \][/tex]
5. Approximate the Roots:
Calculating the approximate values:
[tex]\[ \sqrt{105} \approx 10.246 \][/tex]
Therefore:
[tex]\[ x_1 = \frac{13 + 10.246}{4} \approx \frac{23.246}{4} \approx 5.811 \][/tex]
[tex]\[ x_2 = \frac{13 - 10.246}{4} \approx \frac{2.754}{4} \approx 0.689 \][/tex]
6. Compare with the Given Choices:
Now we compare the approximate roots with the choices:
[tex]\[ \text{Choice (a)}: x = -13.13, \, 3.25 \quad \text{(This does not match our values)} \][/tex]
[tex]\[ \text{Choice (b)}: x = 0.4, \, 0.67 \quad \text{(This does not match our values)} \][/tex]
[tex]\[ \text{Choice (c)}: x = 0.69, \, 5.81 \quad \text{(This closely matches our values: } 0.689 \approx 0.69 \text{ and } 5.811 \approx 5.81) \][/tex]
[tex]\[ \text{Choice (d)}: x = 1.38, \, 11.63 \quad \text{(This does not match our values)} \][/tex]
### Conclusion:
The approximate roots of the equation [tex]\(2 x^2 - 13 x + 8 = 0\)[/tex] are matched correctly by choice (c):
[tex]\[ x = 0.69 \text{ and } 5.81 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{\text{c}} \][/tex]
