To solve the problem of finding the number [tex]\( x \)[/tex] such that the sum of the number and its reciprocal is [tex]\(\frac{10}{3}\)[/tex], let's follow these steps.
1. Form the Equation:
We know that the sum of the number [tex]\( x \)[/tex] and its reciprocal is [tex]\(\frac{10}{3}\)[/tex]. This can be written as:
[tex]\[
x + \frac{1}{x} = \frac{10}{3}
\][/tex]
2. Clear the Fraction:
To simplify the equation, we can multiply every term by [tex]\( x \)[/tex] (assuming [tex]\( x \neq 0 \)[/tex]):
[tex]\[
x^2 + 1 = \frac{10}{3}x
\][/tex]
3. Rearrange into Standard Form:
Next, we need to rearrange the equation into a standard quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex]. To do this, we first eliminate the fraction by multiplying through by 3:
[tex]\[
3x^2 + 3 = 10x
\][/tex]
Now, subtract [tex]\( 10x \)[/tex] from both sides to get:
[tex]\[
3x^2 - 10x + 3 = 0
\][/tex]
4. Solve the Quadratic Equation:
The equation [tex]\( 3x^2 - 10x + 3 = 0 \)[/tex] is a quadratic equation. We can solve it using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
- Here, [tex]\( a = 3 \)[/tex], [tex]\( b = -10 \)[/tex], and [tex]\( c = 3 \)[/tex].
5. Calculate the Discriminant:
First, calculate the discriminant [tex]\( \Delta \)[/tex]:
[tex]\[
\Delta = b^2 - 4ac = (-10)^2 - 4(3)(3) = 100 - 36 = 64
\][/tex]
6. Apply the Quadratic Formula:
Now, use the quadratic formula to find the solutions:
[tex]\[
x = \frac{-(-10) \pm \sqrt{64}}{2 \cdot 3} = \frac{10 \pm 8}{6}
\][/tex]
7. Find the Two Solutions:
- For the positive square root:
[tex]\[
x = \frac{10 + 8}{6} = \frac{18}{6} = 3
\][/tex]
- For the negative square root:
[tex]\[
x = \frac{10 - 8}{6} = \frac{2}{6} = \frac{1}{3}
\][/tex]
Therefore, the numbers that satisfy the given condition [tex]\( x + \frac{1}{x} = \frac{10}{3} \)[/tex] are [tex]\( x = 3 \)[/tex] and [tex]\( x = \frac{1}{3} \)[/tex]. These are the two numbers whose sum with their reciprocal equals [tex]\(\frac{10}{3}\)[/tex].
