Type the correct answer in the box. Use numerals instead of words. What is the solution of this equation?
$-\frac{1}{2} n^2+18=0$
$n= \pm$
Answer (2)
Multiply both sides of the equation by -2: n 2 − 36 = 0
Add 36 to both sides: n 2 = 36
Take the square root of both sides: n = ± 36
Simplify: n = ± 6
Explanation
Understanding the Problem We are given the equation − 2 1 n 2 + 18 = 0 and we want to find the solution for n .
Eliminating the Fraction First, let's multiply both sides of the equation by -2 to get rid of the fraction and the negative sign on the n 2 term: ( − 2 ) × ( − 2 1 n 2 + 18 ) = ( − 2 ) × 0 n 2 − 36 = 0
Isolating the n^2 Term Next, we add 36 to both sides of the equation to isolate the n 2 term: n 2 − 36 + 36 = 0 + 36 n 2 = 36
Taking the Square Root Now, we take the square root of both sides of the equation to solve for n . Remember that when we take the square root, we need to consider both the positive and negative roots: n = ± 36
Simplifying the Solution Finally, we simplify the square root: n = ± 6 So the solutions are n = 6 and n = − 6 .
Examples
Imagine you are designing a square garden with an area of 36 square meters. The side length of the garden can be found by solving the equation s 2 = 36 , where s is the side length. Taking the square root of both sides gives s = ± 6 . Since the side length cannot be negative, we take the positive value, s = 6 meters. This problem demonstrates how quadratic equations can be used to find dimensions in real-world scenarios.
The solutions for the equation − 2 1 n 2 + 18 = 0 are n = 6 and n = − 6 . This shows that both a positive and negative value can satisfy the equation. Thus, the values of n are both important in understanding the equation's solutions.
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