What is factor 3x^{-1/2}+4x^{1/2}+x^{3/2} ?

The solution to factor 3x^{-1/2}+4x^{1/2}+x^{3/2} is ((x+1)(x+3))/(x^{1/2)}

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factor 3x^{-1/2}+4x^{1/2}+x^{3/2}

factor

3 x^{- \frac{1}{2}} + 4 x^{\frac{1}{2}} + x^{\frac{3}{2}}

\frac{( x + 1 ) ( x + 3 )}{x ^{\frac{1}{2}}}

Solution steps

3 x^{- \frac{1}{2}} + 4 x^{\frac{1}{2}} + x^{\frac{3}{2}}

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

x^{\frac{1}{2}} = x^{- \frac{1}{2}} x^{2 \cdot \frac{1}{2}}

= 3 x^{- \frac{1}{2}} + 4 x^{- \frac{1}{2}} x^{2 \cdot \frac{1}{2}} + x^{\frac{3}{2}}

Factor out common term

x^{- \frac{1}{2}}

= x^{- \frac{1}{2}} (3 + 4 x^{2 \cdot \frac{1}{2}}) + x^{\frac{3}{2}}

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

x^{\frac{1}{2}} = x^{1 +- \frac{1}{2}} = x^{1} x^{- \frac{1}{2}}, x^{\frac{3}{2}} = x^{2 +- \frac{1}{2}} = x^{2} x^{- \frac{1}{2}}

= 3 x^{- \frac{1}{2}} + 4 x^{1} x^{- \frac{1}{2}} + x^{2} x^{- \frac{1}{2}}

Factor out common term

x^{- \frac{1}{2}}

= x^{- \frac{1}{2}} (3 + 4 x + x^{2})

Factor

x^{2} + 4 x + 3 : (x + 1) (x + 3)

= x^{- \frac{1}{2}} (x + 1) (x + 3)

Refine

= \frac{( x + 1 ) ( x + 3 )}{x ^{\frac{1}{2}}}

- 3 (x - 4) < 2 (- 2 x - 8) + 12

6 x^{2} + 12 x - 48

x^{2} - 2

- 81 - - 144

2 x - 4 < - 10

What is factor 3x^{-1/2}+4x^{1/2}+x^{3/2} ?
The solution to factor 3x^{-1/2}+4x^{1/2}+x^{3/2} is ((x+1)(x+3))/(x^{1/2)}