Solve:
$\begin{array}{c}
5 x-\frac{16}{x}=-16 \\
x=[?],
\end{array}$
Answer (1)
Multiply both sides by x to get 5 x 2 − 16 = − 16 x .
Rearrange to standard quadratic form: 5 x 2 + 16 x − 16 = 0 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Calculate the solutions: x = 0.8 and x = − 4 , so the final answer is 0.8 , − 4 .
Explanation
Problem Analysis We are given the equation 5 x − x 16 = − 16 . Our goal is to solve for x .
Eliminating the Fraction First, we multiply both sides of the equation by x to eliminate the fraction: x ⋅ ( 5 x − x 16 ) = x ⋅ ( − 16 ) This simplifies to: 5 x 2 − 16 = − 16 x
Rearranging the Equation Next, we rearrange the equation into the standard quadratic form: 5 x 2 + 16 x − 16 = 0
Applying the Quadratic Formula Now, we solve the quadratic equation 5 x 2 + 16 x − 16 = 0 using the quadratic formula: x = 2 a − b ± b 2 − 4 a c where a = 5 , b = 16 , and c = − 16 .
Calculating the Discriminant We calculate the discriminant: b 2 − 4 a c = 1 6 2 − 4 ( 5 ) ( − 16 ) = 256 + 320 = 576
Finding the Square Root of the Discriminant Then, we find the square root of the discriminant: 576 = 24
Calculating Possible Values of x Now, we apply the quadratic formula to find the two possible values of x : x = 2 ( 5 ) − 16 ± 24 = 10 − 16 ± 24
Final Answer Finally, we calculate the two values of x :
x 1 = 10 − 16 + 24 = 10 8 = 5 4 = 0.8 x 2 = 10 − 16 − 24 = 10 − 40 = − 4 Thus, the solutions are x = 0.8 and x = − 4 .
Conclusion Therefore, the solutions to the equation 5 x − x 16 = − 16 are x = 0.8 and x = − 4 .
Examples
Quadratic equations are not just abstract math; they appear in various real-world applications. For instance, when designing a parabolic mirror for a telescope, engineers use quadratic equations to determine the precise curvature needed to focus light correctly. Similarly, in finance, quadratic equations can model investment growth or calculate loan payments, helping individuals and institutions make informed decisions about their money. Understanding how to solve these equations provides valuable tools for problem-solving in science, engineering, and finance.
