$\frac{2 \cdot 5-r}{4}=\frac{1}{r}$
Answer (1)
Simplify the given equation.
Rearrange the equation into the standard quadratic form: r 2 − 10 r + 4 = 0 .
Apply the quadratic formula to solve for r .
The solutions are r = 5 + 21 and r = 5 − 21 , so the final answer is 5 ± 21 .
Explanation
Problem Setup We are given the equation 4 2 × 5 − r = r 1 . Our goal is to solve for r .
Simplify the Equation First, simplify the numerator on the left side: 4 10 − r = r 1 .
Eliminate Fractions Multiply both sides by 4 r to eliminate the fractions: r ( 10 − r ) = 4 .
Expand Expand the left side: 10 r − r 2 = 4 .
Rearrange to Quadratic Form Rearrange the equation into a quadratic form: r 2 − 10 r + 4 = 0 .
Apply Quadratic Formula Use the quadratic formula to solve for r : r = 2 a − b ± b 2 − 4 a c , where a = 1 , b = − 10 , and c = 4 .
Substitute Values Substitute the values of a , b , and c into the quadratic formula: r = 2 ( 1 ) 10 ± ( − 10 ) 2 − 4 ( 1 ) ( 4 ) .
Simplify Simplify the expression: r = 2 10 ± 100 − 16 .
Further Simplify Further simplification: r = 2 10 ± 84 = 2 10 ± 2 21 = 5 ± 21 .
Final Solutions Therefore, the solutions are r = 5 + 21 and r = 5 − 21 .
Examples
Understanding quadratic equations is crucial in various fields, such as physics and engineering. For instance, when calculating the trajectory of a projectile, you often encounter a quadratic equation that describes the height of the object as a function of time. By solving this equation, you can determine when the projectile will hit the ground or reach its maximum height. This principle is applied in designing ballistics, optimizing sports performance, and even in video game physics to create realistic motion.
