Question

an investor sees a transaction summary where the total purchase price before fees was $2,266. the advisor says that, if the share price had been $2.06 cheaper, the investor could have purchased 10 more shares. use the equation \\(\frac{2,266}{x - 2.06} - \frac{2,266}{x} = 10\\) to find the original price per share of the stock. round to the nearest hundredth if necessary. enter your answer in the box. $\square$

Explanation:

Step1: Find common denominator

Multiply terms by $x(x-2.06)$:
$$2266x - 2266(x-2.06) = 10x(x-2.06)$$

Step2: Simplify left side

Expand and cancel like terms:
$$2266x - 2266x + 2266\times2.06 = 10x(x-2.06)$$
$$4667.96 = 10x(x-2.06)$$

Step3: Divide both sides by 10

Simplify the equation:
$$466.796 = x^2 - 2.06x$$
$$x^2 - 2.06x - 466.796 = 0$$

Step4: Apply quadratic formula

Use $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, $a=1,b=-2.06,c=-466.796$:
$$x=\frac{2.06\pm\sqrt{(-2.06)^2-4(1)(-466.796)}}{2(1)}$$
$$x=\frac{2.06\pm\sqrt{4.2436 + 1867.184}}{2}$$
$$x=\frac{2.06\pm\sqrt{1871.4276}}{2}$$
$$x=\frac{2.06\pm43.26}{2}$$

Step5: Select positive solution

Discard negative price:
$$x=\frac{2.06+43.26}{2}=\frac{45.32}{2}=22.66$$

Answer:

$\$22.66$