To solve the quadratic equation [tex]\( 3x^2 - 10x + 5 = 0 \)[/tex] and round the solutions to the nearest hundredth, we can follow these steps:
1. Identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] from the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = -10\)[/tex]
- [tex]\(c = 5\)[/tex]
2. Calculate the discriminant [tex]\( \Delta \)[/tex] using the formula [tex]\( \Delta = b^2 - 4ac \)[/tex]:
[tex]\[
\Delta = (-10)^2 - 4 \cdot 3 \cdot 5 = 100 - 60 = 40
\][/tex]
3. Find the two solutions [tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex] using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{\Delta}}{2a} \)[/tex]:
[tex]\[
x_1 = \frac{-(-10) + \sqrt{40}}{2 \cdot 3} = \frac{10 + \sqrt{40}}{6}
\][/tex]
[tex]\[
x_2 = \frac{-(-10) - \sqrt{40}}{2 \cdot 3} = \frac{10 - \sqrt{40}}{6}
\][/tex]
4. Calculate the approximate values of the solutions:
- For [tex]\(x_1\)[/tex]:
[tex]\[
x_1 \approx \frac{10 + 6.3246}{6} \approx \frac{16.3246}{6} \approx 2.720759220056127
\][/tex]
- For [tex]\(x_2\)[/tex]:
[tex]\[
x_2 \approx \frac{10 - 6.3246}{6} \approx \frac{3.6754}{6} \approx 0.6125741132772068
\][/tex]
5. Round the solutions to the nearest hundredth:
- For [tex]\(x_1\)[/tex], rounding 2.720759220056127 to the nearest hundredth gives:
[tex]\[
x_1 \approx 2.72
\][/tex]
- For [tex]\(x_2\)[/tex], rounding 0.6125741132772068 to the nearest hundredth gives:
[tex]\[
x_2 \approx 0.61
\][/tex]
Therefore, the solutions to the quadratic equation [tex]\( 3x^2 - 10x + 5 = 0 \)[/tex], rounded to the nearest hundredth, are [tex]\( x_1 \approx 2.72 \)[/tex] and [tex]\( x_2 \approx 0.61 \)[/tex].
