Answer :
1/2s=4s-21 is s = 5.275
To solve the equation [tex]\( \frac{1}{2s} = 4s - 21 \)[/tex], follow these steps:
1. Multiply Both Sides by 2s: This step helps eliminate the fraction.
[tex]\[ 2s \times \frac{1}{2s} = (4s - 21) \times 2s \][/tex]
Simplify the left side to just 1, and distribute 2s on the right side:
[tex]\[ 1 = 8s^2 - 42s \][/tex]
2. Rearrange and Set to Zero: Move all terms to one side of the equation to solve for ( s ).
[tex]\[ 8s^2 - 42s - 1 = 0 \][/tex]
Now, to solve the quadratic equation[tex]\( 8s^2 - 42s - 1 = 0 \),[/tex]you can use the quadratic formula:
[tex]\[ s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
In this equation, ( a = 8 ), ( b = -42), and ( c = -1 ). Plug these values into the formula to find the solutions for ( s ):
[tex]\[ s = \frac{42 \pm \sqrt{(-42)^2 - 4(8)(-1)}}{2(8)} \][/tex]
[tex]\[ s = \frac{42 \pm \sqrt{1764 + 32}}{16} \][/tex]
[tex]\[ s = \frac{42 \pm \sqrt{1796}}{16} \][/tex]
[tex]\[ s = \frac{42 \pm 42.41}{16} \][/tex]
Now, calculate both possible solutions:
[tex]\[ s_1 = \frac{42 + 42.41}{16} \approx \frac{84.41}{16} \approx 5.275 \][/tex]
[tex]\[ s_2 = \frac{42 - 42.41}{16} \approx \frac{-0.41}{16} \approx -0.0256 \][/tex]
The final answers are approximately[tex]\( s_1 = 5.275 \) and \( s_2 = -0.0256 \).[/tex]
In this case, only the positive solution makes sense in the context of the original equation, as ( s ) is likely representing a positive quantity. Thus, ( s = 5.275 ) is the solution.
By multiplying both sides of the equation by ( 2s ), we eliminated the fraction, leading to a quadratic equation. Using the quadratic formula, we solved for ( s ) by substituting the coefficients ( a ), ( b ), and ( c ). The final solutions were then calculated and checked for validity in the original equation, resulting in ( s = 5.275).
complete question
What is 1/2s=4s-21.
