To solve the quadratic equation [tex]\(3x^2 - 7x + 4 = 0\)[/tex] using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
we need to identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] first:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = -7\)[/tex]
- [tex]\(c = 4\)[/tex]
Next, we calculate the discriminant:
[tex]\[ \text{Discriminant} = b^2 - 4ac \][/tex]
[tex]\[ \text{Discriminant} = (-7)^2 - 4(3)(4) \][/tex]
[tex]\[ \text{Discriminant} = 49 - 48 \][/tex]
[tex]\[ \text{Discriminant} = 1 \][/tex]
Since the discriminant is positive, we will have two real solutions. Let's calculate these solutions using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\text{Discriminant}}}{2a} \][/tex]
Substituting in the values for [tex]\(a\)[/tex], [tex]\(b\)[/tex], and the discriminant:
[tex]\[ x = \frac{-(-7) \pm \sqrt{1}}{2(3)} \][/tex]
[tex]\[ x = \frac{7 \pm 1}{6} \][/tex]
We now find the two solutions:
For the positive sign:
[tex]\[ x_1 = \frac{7 + 1}{6} = \frac{8}{6} = \frac{4}{3} \approx 1.3333333333333333 \][/tex]
For the negative sign:
[tex]\[ x_2 = \frac{7 - 1}{6} = \frac{6}{6} = 1 \][/tex]
Therefore, the solutions to the equation [tex]\(3x^2 - 7x + 4 = 0\)[/tex] are:
[tex]\[ x = \frac{4}{3}, 1 \][/tex]
Or in decimal form:
[tex]\[ x \approx 1.3333333333333333, 1 \][/tex]
Thus, the correct answer is:
[tex]\[
1.3333333333333333, 1
\][/tex]
