To find the solutions for the quadratic equation [tex]\(3d^2 - 8d + 3 = 0\)[/tex], we use the quadratic formula, which is given by:
[tex]\[
d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the coefficients of the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]. For our equation [tex]\(3d^2 - 8d + 3 = 0\)[/tex]:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = -8\)[/tex]
- [tex]\(c = 3\)[/tex]
1. Calculate the discriminant, which is [tex]\(b^2 - 4ac\)[/tex]:
[tex]\[
\text{Discriminant} = (-8)^2 - 4 \cdot 3 \cdot 3 = 64 - 36 = 28
\][/tex]
2. Substitute the values into the quadratic formula:
[tex]\[
d = \frac{-(-8) \pm \sqrt{28}}{2 \cdot 3} = \frac{8 \pm \sqrt{28}}{6}
\][/tex]
3. Simplify the expressions:
- The solution with the positive square root:
[tex]\[
d_1 = \frac{8 + \sqrt{28}}{6}
\][/tex]
- The solution with the negative square root:
[tex]\[
d_2 = \frac{8 - \sqrt{28}}{6}
\][/tex]
4. Approximate the square root of 28:
[tex]\[
\sqrt{28} \approx 5.29
\][/tex]
5. Substitute this approximation back into the equations:
[tex]\[
d_1 = \frac{8 + 5.29}{6} \approx \frac{13.29}{6} \approx 2.215
\][/tex]
[tex]\[
d_2 = \frac{8 - 5.29}{6} \approx \frac{2.71}{6} \approx 0.452
\][/tex]
6. Round the solutions to the nearest tenth:
[tex]\[
d_1 \approx 2.2
\][/tex]
[tex]\[
d_2 \approx 0.5
\][/tex]
Thus, the solutions to the equation [tex]\(3d^2 - 8d + 3 = 0\)[/tex], rounded to the nearest tenth, are [tex]\(d = 2.2\)[/tex] and [tex]\(d = 0.5\)[/tex].
