To solve the quadratic equation [tex]\(3d^2 - 8d + 3 = 0\)[/tex] for all values of [tex]\(d\)[/tex], we can use the quadratic formula:
[tex]\[ d = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\(a = 3\)[/tex], [tex]\(b = -8\)[/tex], and [tex]\(c = 3\)[/tex].
Step 1: Calculate the Discriminant
First, we need to calculate the discriminant, [tex]\(\Delta\)[/tex]:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = (-8)^2 - 4 \cdot 3 \cdot 3 = 64 - 36 = 28
\][/tex]
Step 2: Calculate the Two Possible Solutions
We now have the discriminant, [tex]\(\Delta = 28\)[/tex]. We can proceed to find the two possible solutions for [tex]\(d\)[/tex]:
[tex]\[
d = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substitute the values:
[tex]\[
d = \frac{-(-8) \pm \sqrt{28}}{2 \cdot 3} = \frac{8 \pm \sqrt{28}}{6}
\][/tex]
This gives us two solutions:
[tex]\[
d_1 = \frac{8 + \sqrt{28}}{6}
\][/tex]
[tex]\[
d_2 = \frac{8 - \sqrt{28}}{6}
\][/tex]
Step 3: Calculate Each Root
- For [tex]\(d_1\)[/tex]:
[tex]\[
d_1 = \frac{8 + \sqrt{28}}{6} \approx \frac{8 + 5.291502622129181}{6} \approx \frac{13.291502622129181}{6} \approx 2.2152504370215302
\][/tex]
- For [tex]\(d_2\)[/tex]:
[tex]\[
d_2 = \frac{8 - \sqrt{28}}{6} \approx \frac{8 - 5.291502622129181}{6} \approx \frac{2.708497377870819}{6} \approx 0.45141622964513645
\][/tex]
Step 4: Rounding to the Nearest Tenth
Finally, we round each solution to the nearest tenth:
[tex]\[
d_1 \approx 2.2
\][/tex]
[tex]\[
d_2 \approx 0.5
\][/tex]
Conclusion
The solutions to the quadratic equation [tex]\(3d^2 - 8d + 3 = 0\)[/tex], rounded to the nearest tenth, are:
[tex]\[
d_1 \approx 2.2
\][/tex]
[tex]\[
d_2 \approx 0.5
\][/tex]
