What is expand (a*x+b*x^{1/2}+c)^2 ?

The solution to expand (a*x+b*x^{1/2}+c)^2 is c^2+a^2x^2+2ax^{3/2}b+xb^2+2acx+2x^{1/2}bc

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expand (ax+bx^{1/2}+c)^2

expand

(a \cdot x + b \cdot x^{\frac{1}{2}} + c)^{2}

c^{2} + a^{2} x^{2} + 2 a x^{\frac{3}{2}} b + x b^{2} + 2 a c x + 2 x^{\frac{1}{2}} b c

Solution steps

(a x + b x^{\frac{1}{2}} + c)^{2}

Apply exponent rule:

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

(a x + b x^{\frac{1}{2}} + c)^{2} = (a x + b x^{\frac{1}{2}} + c) (a x + b x^{\frac{1}{2}} + c)

= (a x + b x^{\frac{1}{2}} + c) (a x + b x^{\frac{1}{2}} + c)

Distribute parentheses

= a x a x + a x b x^{\frac{1}{2}} + a x c + b x^{\frac{1}{2}} a x + b x^{\frac{1}{2}} b x^{\frac{1}{2}} + b x^{\frac{1}{2}} c + c a x + c b x^{\frac{1}{2}} + cc

Simplify

a x a x + a x b x^{\frac{1}{2}} + a x c + b x^{\frac{1}{2}} a x + b x^{\frac{1}{2}} b x^{\frac{1}{2}} + b x^{\frac{1}{2}} c + c a x + c b x^{\frac{1}{2}} + cc : c^{2} + a^{2} x^{2} + 2 a x^{\frac{3}{2}} b + x b^{2} + 2 a c x + 2 x^{\frac{1}{2}} b c

= c^{2} + a^{2} x^{2} + 2 a x^{\frac{3}{2}} b + x b^{2} + 2 a c x + 2 x^{\frac{1}{2}} b c

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What is expand (a*x+b*x^{1/2}+c)^2 ?
The solution to expand (a*x+b*x^{1/2}+c)^2 is c^2+a^2x^2+2ax^{3/2}b+xb^2+2acx+2x^{1/2}bc