find a simplified form for ( f(x) ). assume ( x ) can be any real number, and write your answer in radical…

find a simplified form for ( f(x) ). assume ( x ) can be any real number, and write your answer in radical form.\n\n( f(x)=sqrt{5 x^{2}-30 x + 45} )\n\n( f(x)=square )

find a simplified form for ( f(x) ). assume ( x ) can be any real number, and write your answer in radical form.\n\n( f(x)=sqrt{5 x^{2}-30 x + 45} )\n\n( f(x)=square )

Answer

Explanation:

Step1: Factor out the common factor

Factor out (5) from the expression inside the square root: [ \begin{align*} f(x)&=\sqrt{5x^{2}-30x + 45}\ &=\sqrt{5(x^{2}-6x + 9)} \end{align*} ]

Step2: Recognize the perfect - square trinomial

Since (x^{2}-6x + 9=(x - 3)^{2}) (using the formula ((a - b)^{2}=a^{2}-2ab + b^{2}) where (a=x) and (b = 3)), then (f(x)=\sqrt{5(x - 3)^{2}})

Step3: Simplify the square - root

Using the property (\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}) ((a = 5), (b=(x - 3)^{2})) and (\sqrt{a^{2}}=\vert a\vert), we get (f(x)=\sqrt{5}\vert x - 3\vert)

Answer:

(\sqrt{5}\vert x - 3\vert)