Final answer:
The values of x as decimals to three significant figures are approximately 0.541 and -5.541.
Step-by-step explanation:
To solve the equation 3/x + 5 = x, we need to find values of x that make the equation true. Since x is in the denominator on the left side, we must treat this as a quadratic equation.
- First, bring all terms to one side to set the equation to zero: 3/x + 5 - x = 0.
- Multiply every term by x to remove the fraction: 3 + 5x - x^2 = 0.
- Rewrite in standard quadratic form: -x^2 + 5x + 3 = 0.
- Since factoring might be complicated, use the quadratic formula: x = [-b ± sqrt(b^2-4ac)]/(2a).
- For our equation, a = -1, b = 5, and c = 3.
- Substitute these values: x = [-5 ± sqrt(5^2-4(-1)(3))]/(2(-1)).
- Calculate the discriminant: sqrt(25 + 12) = sqrt(37).
- Compute both solutions: x = (5 ± sqrt(37)) / -2.
- Round each solution to three decimal places. x ≈ 0.541 and x ≈ -5.541.
Thus, the two decimal values for x that satisfy the original equation 3/x + 5 = x to three significant figures are approximately 0.541 and -5.541.