Solving Quadratic Equation using the Formula

The quadratic equation factors into $(2x + 7)(3x - 5) = 0 \Rightarrow x = -\frac{7}{2}, \frac{5}{3}$ , or $x = \boxed{-3.5, 1.7}.$

Use Vieta's formulas: the sum of the roots is $-\frac{11}{6} \approx -1.83$ and the product of the roots is $-\frac{35}{6} \approx -5.83$ . In these options, $x = (1.7, -3.5)$ matches these values most closely: the sum is $-1.8$ and the product is $-5.95$ .

If the values were changed by one decimal place, such as: $x = (1.6, -3.4)$ , the product of the roots would differ from the original pair by $-0.51$ . This error is too large, so this cannot be the answer. The error is too large for the other three possible pairs: $x = (1.6, -3.6)$ , $(1.8, -3.4)$ , and $(1.8, -3.6)$ .

We can use the quadratic formula

$x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$

Substitute:

$x=\dfrac{-11 \pm \sqrt{11^2-4(6)(-35)}}{2(6)}=\dfrac{-11}{12} \pm \dfrac{31}{12}$

$x_1=\dfrac{-11+31}{12} \approx 1.7$

$x_2=\dfrac{-11-31}{12} \approx -3.5$

The desired answer is $1.7$ and $-3.5$ .