Question

im2 midterm study guide #3
question 8
solve the equation $n = \sqrt{13n - 40}$
answer
additional solution no solution
$n = $

Explanation:

Step1: Square both sides

To eliminate the square root, square both sides of the equation \( n = \sqrt{13n - 40} \). This gives \( n^2 = (\sqrt{13n - 40})^2 \), which simplifies to \( n^2 = 13n - 40 \).

Step2: Rearrange into quadratic form

Rearrange the equation \( n^2 = 13n - 40 \) to standard quadratic form \( ax^2 + bx + c = 0 \). Subtract \( 13n \) and add \( 40 \) to both sides: \( n^2 - 13n + 40 = 0 \).

Step3: Factor the quadratic

Factor the quadratic equation \( n^2 - 13n + 40 = 0 \). We need two numbers that multiply to \( 40 \) and add to \( -13 \). These numbers are \( -5 \) and \( -8 \), so the equation factors as \( (n - 5)(n - 8) = 0 \).

Step4: Solve for n

Set each factor equal to zero and solve:

• For \( n - 5 = 0 \), we get \( n = 5 \).
• For \( n - 8 = 0 \), we get \( n = 8 \).

Step5: Check solutions

Check each solution in the original equation \( n = \sqrt{13n - 40} \):

• For \( n = 5 \): Left side is \( 5 \), right side is \( \sqrt{13(5) - 40} = \sqrt{65 - 40} = \sqrt{25} = 5 \). So \( n = 5 \) is valid.
• For \( n = 8 \): Left side is \( 8 \), right side is \( \sqrt{13(8) - 40} = \sqrt{104 - 40} = \sqrt{64} = 8 \). So \( n = 8 \) is valid.

Answer:

\( n = 5 \) or \( n = 8 \) (both solutions are valid)