Question

which equation has the solutions ( x = \frac{5 pm 2sqrt{7}}{3} )?

• ( 3x^2 - 5x + 7 = 0 )
• ( 3x^2 - 5x - 1 = 0 )
• ( 3x^2 - 10x + 6 = 0 )
• ( 3x^2 - 10x - 1 = 0 )

Explanation:

Step1: Recall quadratic formula

For $ax^2+bx+c=0$, $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Step2: Match given solution to formula

Given $x=\frac{5\pm2\sqrt{7}}{3}=\frac{5\pm\sqrt{28}}{3}$. Rewrite denominator as $2a$: $2a=3\Rightarrow a=\frac{3}{2}$, but multiply numerator/denominator by 2: $x=\frac{10\pm\sqrt{112}}{6}$. Now $2a=6\Rightarrow a=3$, $-b=10\Rightarrow b=-10$.

Step3: Calculate discriminant

$\sqrt{b^2-4ac}=\sqrt{112}$, so $b^2-4ac=112$. Substitute $a=3,b=-10$:
$(-10)^2-4(3)c=112$
$100-12c=112$

Step4: Solve for c

$-12c=112-100=12\Rightarrow c=-1$

Step5: Form quadratic equation

Substitute $a=3,b=-10,c=-1$: $3x^2-10x-1=0$

Answer:

$3x^2 - 10x - 1 = 0$ (the last option)