To find the exact value of x, we need to use the definition of the golden ratio. The golden ratio states that the ratio of a larger quantity (x) to a smaller quantity (10) is equal to the ratio of the sum of the two quantities (x + 10) to the larger quantity (x).
Mathematically, this can be expressed as:
x / 10 = (x + 10) / x
To solve this equation, we can start by cross-multiplying:
x * x = 10 * (x + 10)
Simplifying further:
x^2 = 10x + 100
Rearranging the equation:
x^2 - 10x - 100 = 0
Now, we have a quadratic equation. To solve it, we can use the quadratic formula:
x = (-b Β± β(b^2 - 4ac)) / (2a)
For our equation, a = 1, b = -10, and c = -100. Substituting these values into the formula:
x = (-(-10) Β± β((-10)^2 - 4 * 1 * -100)) / (2 * 1)
= (10 Β± β(100 + 400)) / 2
= (10 Β± β500) / 2
To simplify the radical, we can break down 500 into its factors:
x = (10 Β± β(100 * 5)) / 2
= (10 Β± β100 * β5) / 2
= (10 Β± 10β5) / 2
= 10/2 Β± 10β5/2
= 5 Β± 5β5
Therefore, the two possible values of x in the golden ratio with 10 (where x is larger than 10) are:
x = 5 + 5β5
or
x = 5 - 5β5