Question

question 39 of 41
which of the following is equivalent to the radical expression below when ( x geq 2 )?
( sqrt{x - 2} \bullet sqrt{x + 9} )

a. ( sqrt{x^2 - 11x - 18} )
b. ( sqrt{x^2 + 7x - 18} )
c. ( sqrt{x^2 - 7x - 18} )
d. ( sqrt{x^2 + 11x - 18} )

Explanation:

Step1: Recall the property of radicals

The property of radicals states that \(\sqrt{a} \cdot \sqrt{b}=\sqrt{ab}\) for \(a\geq0\) and \(b\geq0\). Here, since \(x\geq2\), \(x - 2\geq0\) and \(x + 9\geq0\) (because when \(x\geq2\), \(x+9\) is definitely positive). So we can apply this property.

Step2: Multiply the expressions inside the radicals

Using the property \(\sqrt{x - 2}\cdot\sqrt{x + 9}=\sqrt{(x - 2)(x + 9)}\). Now we expand \((x - 2)(x + 9)\) using the distributive property (FOIL method).
\((x-2)(x + 9)=x\cdot x+x\cdot9-2\cdot x-2\cdot9=x^{2}+9x-2x - 18=x^{2}+7x-18\).
So \(\sqrt{(x - 2)(x + 9)}=\sqrt{x^{2}+7x - 18}\).

Answer:

B. \(\sqrt{x^{2}+7x - 18}\)