Question

david took out a $70,000 loan from his bank in 2010. the quadratic equation that approximates the year/loan balance relationship is $y = -98x^2 - 1,531x + 69,718$, where $x$ is the number of years and $y$ is the balance. in what year will his loan balance be $37,234?

a 2022

b 2023

c 2024

d 2025

Explanation:

Step1: Set $y=37234$, rearrange equation

$37234 = -98x^2 - 1531x + 69718$
$98x^2 + 1531x - 32484 = 0$

Step2: Apply quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Here $a=98$, $b=1531$, $c=-32484$
First calculate discriminant:
$\Delta = 1531^2 - 4\times98\times(-32484) = 2343961 + 12709632 = 15053593$
$\sqrt{\Delta} \approx 3880$
$x = \frac{-1531 + 3880}{2\times98} = \frac{2349}{196} \approx 12.0$

Step3: Calculate target year

$2010 + 12 = 2022$

Answer:

A 2022