To determine which equation has the solutions [tex]\( x = \frac{5 \pm 2 \sqrt{7}}{3} \)[/tex], we need to compare each of the given quadratic equations to see which one has these solutions. Let's go through each equation step-by-step:
1. Equation: [tex]\(3x^2 - 5x + 7 = 0\)[/tex]:
- Using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\)[/tex], where [tex]\(a = 3\)[/tex], [tex]\(b = -5\)[/tex], and [tex]\(c = 7\)[/tex]:
- Calculate the discriminant: [tex]\(b^2 - 4ac = (-5)^2 - 4 \cdot 3 \cdot 7 = 25 - 84 = -59\)[/tex].
- The discriminant is negative, meaning the equation has complex roots, not real roots. Hence, this equation does not have the given solutions.
2. Equation: [tex]\(3x^2 - 5x - 1 = 0\)[/tex]:
- Using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\)[/tex], where [tex]\(a = 3\)[/tex], [tex]\(b = -5\)[/tex], and [tex]\(c = -1\)[/tex]:
- Calculate the discriminant: [tex]\(b^2 - 4ac = (-5)^2 - 4 \cdot 3 \cdot (-1) = 25 + 12 = 37\)[/tex].
- Calculate the roots:
[tex]\[
x = \frac{5 \pm \sqrt{37}}{6}
\][/tex]
- The solutions are not in the form [tex]\( \frac{5 \pm 2\sqrt{7}}{3} \)[/tex]. Hence, this equation does not have the given solutions.
3. Equation: [tex]\(3x^2 - 10x + 6 = 0\)[/tex]:
- Using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\)[/tex], where [tex]\(a = 3\)[/tex], [tex]\(b = -10\)[/tex], and [tex]\(c = 6\)[/tex]:
- Calculate the discriminant: [tex]\(b^2 - 4ac = (-10)^2 - 4 \cdot 3 \cdot 6 = 100 - 72 = 28\)[/tex].
- Calculate the roots:
[tex]\[
x = \frac{10 \pm \sqrt{28}}{6} = \frac{10 \pm 2\sqrt{7}}{6} = \frac{5 \pm \sqrt{7}}{3}
\][/tex]
- This expression is not equal to [tex]\( \frac{5 \pm 2\sqrt{7}}{3} \)[/tex]. Hence, this equation does not have the given solutions.
4. Equation: [tex]\(3x^2 - 10x - 1 = 0\)[/tex]:
- Using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\)[/tex], where [tex]\(a = 3\)[/tex], [tex]\(b = -10\)[/tex], and [tex]\(c = -1\)[/tex]:
- Calculate the discriminant: [tex]\(b^2 - 4ac = (-10)^2 - 4 \cdot 3 \cdot (-1) = 100 + 12 = 112\)[/tex].
- Calculate the roots:
[tex]\[
x = \frac{10 \pm \sqrt{112}}{6} = \frac{10 \pm 4\sqrt{7}}{6} = \frac{5 \pm 2\sqrt{7}}{3}
\][/tex]
- These solutions match exactly with [tex]\( \frac{5 \pm 2 \sqrt{7}}{3} \)[/tex]. Thus, this equation has the given solutions.
So, the equation [tex]\(3x^2 - 10x - 1 = 0\)[/tex] is the one that has the solutions [tex]\( x = \frac{5 \pm 2 \sqrt{7}}{3} \)[/tex].
