Solve the equation for [tex]$s$[/tex].
[tex]$64 s^2+80 s=-25$[/tex]
Answer (2)
Rewrite the equation in standard quadratic form: 64 s 2 + 80 s + 25 = 0 .
Factor the quadratic expression: ( 8 s + 5 ) 2 = 0 .
Solve for s : 8 s + 5 = 0 ⟹ s = 8 − 5 .
The solution to the equation is − 8 5 .
Explanation
Problem Analysis We are given the quadratic equation 64 s 2 + 80 s = − 25 . Our goal is to solve for s .
Rewrite in Standard Form First, we rewrite the equation in the standard quadratic form a x 2 + b x + c = 0 . Adding 25 to both sides, we get: 64 s 2 + 80 s + 25 = 0
Factor the Quadratic Now, we can solve this quadratic equation. We can try to factor the quadratic expression. Notice that 64 s 2 = ( 8 s ) 2 , 25 = 5 2 , and 80 s = 2 ( 8 s ) ( 5 ) . Thus, the quadratic expression is a perfect square: ( 8 s + 5 ) 2 = 0
Take the Square Root Taking the square root of both sides, we get: 8 s + 5 = 0
Isolate s Solving for s , we subtract 5 from both sides: 8 s = − 5
Solve for s Finally, we divide by 8: s = 8 − 5 = − 0.625
Final Answer Therefore, the solution to the equation is s = − 0.625 .
Examples
Quadratic equations are incredibly useful in various real-life scenarios. For instance, they can model the trajectory of a ball thrown in the air, helping to determine its maximum height and range. In business, quadratic equations can be used to analyze profit margins, optimizing production levels to maximize revenue. Understanding how to solve these equations provides valuable insights in physics, engineering, economics, and many other fields.
To solve the equation 64 s 2 + 80 s + 25 = 0 , we factor it as ( 8 s + 5 ) 2 = 0 and find that s = − 8 5 or − 0.625 .
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