Question

question #3
a rectangle has a perimeter of 30 in, length of 6 in. and a width of \\( \sqrt { x + 2 } \\) in. use this information to find the actual width.
o 18 inches
o 79 inches
o 12 inches
o 9 inches

question #4
solve the radical equation. check for extraneous solutions \\( - 3 = - n + \sqrt { 5 n - 9 } \\)
\\( \square n = 9 \\)
\\( \square n = 2 ; n = - 9 \\)
\\( \square n = - 4 ; n = - 9 \\)
\\( \square n = 2 \\)

Explanation:

Step1: Use the perimeter formula for a rectangle

The perimeter formula of a rectangle is \(P = 2(l + w)\), where \(P\) is the perimeter, \(l\) is the length, and \(w\) is the width. Given \(P=30\) in and \(l = 6\) in, and \(w=\sqrt{x + 2}\) in. Substitute into the formula: \(30=2(6+\sqrt{x + 2})\).
Divide both sides by 2: \(\frac{30}{2}=6+\sqrt{x + 2}\), so \(15=6+\sqrt{x + 2}\).

Step2: Solve for \(\sqrt{x + 2}\)

Subtract 6 from both sides: \(15 - 6=\sqrt{x + 2}\), so \(\sqrt{x + 2}=9\).

Step1: Isolate the radical term

Given \(-3=-n+\sqrt{5n - 9}\), add \(n\) to both sides: \(n - 3=\sqrt{5n - 9}\).

Step2: Square both sides

\((n - 3)^2=(\sqrt{5n - 9})^2\). Expand the left - hand side using \((a - b)^2=a^2-2ab + b^2\), so \(n^2-6n + 9=5n - 9\).

Step3: Rearrange to form a quadratic equation

Move all terms to one side: \(n^2-6n-5n + 9 + 9=0\), which simplifies to \(n^2-11n + 18=0\).
Factor the quadratic equation: \(n^2-11n + 18=(n - 2)(n - 9)=0\). So \(n=2\) or \(n = 9\).

Step4: Check for extraneous solutions

When \(n = 2\):
Left - hand side of the original equation: \(-3\).
Right - hand side: \(-2+\sqrt{5\times2-9}=-2+\sqrt{10 - 9}=-2 + 1=-1
eq-3\).
When \(n = 9\):
Left - hand side of the original equation: \(-3\).
Right - hand side: \(-9+\sqrt{5\times9-9}=-9+\sqrt{45 - 9}=-9+\sqrt{36}=-9 + 6=-3\).

Answer:

9 inches