Sure, let's determine which equation has the solutions [tex]\( x = \frac{5 \pm 2 \sqrt{7}}{3} \)[/tex].
We are given four possible quadratic equations, and our goal is to identify which one has the given solutions. We start by using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]. We will compare the forms to find out which set of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] match our solutions.
First, note that the general form of a quadratic equation is:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
Given the solutions [tex]\( x = \frac{5 \pm 2 \sqrt{7}}{3} \)[/tex], rewrite it in the form of the quadratic formula [tex]\(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex].
From [tex]\( x = \frac{5 \pm 2 \sqrt{7}}{3} \)[/tex], we can identify:
[tex]\[ a = 3 \][/tex]
[tex]\[ b = -5 \][/tex]
(to make [tex]\( \frac{5}{3} \)[/tex] as the middle term of the quadratic formula)
[tex]\[ \text{Discriminant: } b^2 - 4ac = (5)^2 - (2 \cdot 3)(2 \sqrt{7})^2 = 25 - 4ac \][/tex]
[tex]\[ 2 \sqrt{7} implies that 4ac needs to be 4 \cdot 3 \cdot 7 = 84 \Rightarrow c = 7\][/tex]
Thus, our equation, substituting in [tex]\( a = 3 \)[/tex], [tex]\( b = -5\)[/tex], and [tex]\( c = 7\)[/tex] as follows:
[tex]\[ ax^2 - bx + c = 0 \][/tex]
Which indicates:
[tex]\[ 3x^2 - 5x + 7 = 0 \][/tex]
Now, let's verify from the given options:
(A) [tex]\( 3x^2 - 5x + 7 = 0 \)[/tex]
(B) [tex]\( 3x^2 - 5x - 1 = 0 \)[/tex]
(C) [tex]\( 3x^2 - 10x + 6 = 0 \)[/tex]
(D) [tex]\( 3x^2 - 10x - 1 = 0 \)[/tex]
The correct equation that aligns with the given solutions is:
[tex]\[ 3x^2 - 5x + 7 = 0 \][/tex]
Therefore, the answer is:
(A) [tex]\( 3 x^2 - 5 x + 7 = 0 \)[/tex]
