Question

solving equations with the quadratic formula
a \golden rectangle\ is a rectangle where the ratio of the longer side to the shorter side is the \golden ratio.\ these rectangles are said to be visually pleasing. an example of a \golden rectangle\ has a length equal to x units and a width equal to x - 1 units. its area is 1 square unit. what is the length of this golden rectangle?
\\( \frac { - 1 + \sqrt { 5 } } { 2 } \\)
\\( \frac { 1 + \sqrt { 5 } } { 2 } \\)
\\( \frac { 1 + \sqrt { 3 } } { 2 } \\)
\\( \frac { 1 - \sqrt { 5 } } { 2 } \\)

Explanation:

Step1: Set up the equation

The area of a rectangle is \(A = length\times width\). Given length \(x\), width \(x - 1\), and \(A=1\). So, \(x(x - 1)=1\).
Expand the left - hand side: \(x^{2}-x = 1\).
Rearrange to the standard quadratic form \(ax^{2}+bx + c = 0\): \(x^{2}-x - 1=0\).
Here, \(a = 1\), \(b=-1\), \(c = - 1\).

Step2: Apply the quadratic formula

The quadratic formula is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).
Substitute \(a = 1\), \(b=-1\), \(c = - 1\) into the formula:

$$
LATEXBLOCK0
$$

Since \(x\) represents the length of a side of a rectangle, \(x>0\).
\(\frac{1-\sqrt{5}}{2}\approx\frac{1 - 2.24}{2}<0\) (rejected as length cannot be negative).
\(\frac{1+\sqrt{5}}{2}\approx\frac{1+2.24}{2}>0\)

Answer:

\(\frac{1+\sqrt{5}}{2}\) (corresponds to the second option \(\frac{1+\sqrt{5}}{2}\))