Solve -4frac{1/2x*sqrt{5}+9}{(5sqrt{5}+1)^2}

$$\frac { - 4 \frac { 1 } { 2 } x \cdot \sqrt { 5 } + 9 } { ( 5 \sqrt { 5 } + 1 ) ^ { 2 } }$$

Factor

$$\frac{9\left(63-5\sqrt{5}\right)\left(-\sqrt{5}x+2\right)}{15376}$$

Solution Steps

Multiply 4 and 2 to get 8.

$$factor(\frac{\left(-\frac{8+1}{2}\right)x\sqrt{5}+9}{\left(5\sqrt{5}+1\right)^{2}})$$

Add 8 and 1 to get 9.

$$factor(\frac{-\frac{9}{2}x\sqrt{5}+9}{\left(5\sqrt{5}+1\right)^{2}})$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5\sqrt{5}+1\right)^{2}.

$$factor(\frac{-\frac{9}{2}x\sqrt{5}+9}{25\left(\sqrt{5}\right)^{2}+10\sqrt{5}+1})$$

The square of \sqrt{5} is 5.

$$factor(\frac{-\frac{9}{2}x\sqrt{5}+9}{25\times 5+10\sqrt{5}+1})$$

Multiply 25 and 5 to get 125.

$$factor(\frac{-\frac{9}{2}x\sqrt{5}+9}{125+10\sqrt{5}+1})$$

Add 125 and 1 to get 126.

$$factor(\frac{-\frac{9}{2}x\sqrt{5}+9}{126+10\sqrt{5}})$$

Rationalize the denominator of \frac{-\frac{9}{2}x\sqrt{5}+9}{126+10\sqrt{5}} by multiplying numerator and denominator by 126-10\sqrt{5}.

$$factor(\frac{\left(-\frac{9}{2}x\sqrt{5}+9\right)\left(126-10\sqrt{5}\right)}{\left(126+10\sqrt{5}\right)\left(126-10\sqrt{5}\right)})$$

Consider \left(126+10\sqrt{5}\right)\left(126-10\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$factor(\frac{\left(-\frac{9}{2}x\sqrt{5}+9\right)\left(126-10\sqrt{5}\right)}{126^{2}-\left(10\sqrt{5}\right)^{2}})$$

Calculate 126 to the power of 2 and get 15876.

$$factor(\frac{\left(-\frac{9}{2}x\sqrt{5}+9\right)\left(126-10\sqrt{5}\right)}{15876-\left(10\sqrt{5}\right)^{2}})$$

Expand \left(10\sqrt{5}\right)^{2}.

$$factor(\frac{\left(-\frac{9}{2}x\sqrt{5}+9\right)\left(126-10\sqrt{5}\right)}{15876-10^{2}\left(\sqrt{5}\right)^{2}})$$

Calculate 10 to the power of 2 and get 100.

$$factor(\frac{\left(-\frac{9}{2}x\sqrt{5}+9\right)\left(126-10\sqrt{5}\right)}{15876-100\left(\sqrt{5}\right)^{2}})$$

The square of \sqrt{5} is 5.

$$factor(\frac{\left(-\frac{9}{2}x\sqrt{5}+9\right)\left(126-10\sqrt{5}\right)}{15876-100\times 5})$$

Multiply 100 and 5 to get 500.

$$factor(\frac{\left(-\frac{9}{2}x\sqrt{5}+9\right)\left(126-10\sqrt{5}\right)}{15876-500})$$

Subtract 500 from 15876 to get 15376.

$$factor(\frac{\left(-\frac{9}{2}x\sqrt{5}+9\right)\left(126-10\sqrt{5}\right)}{15376})$$

Use the distributive property to multiply -\frac{9}{2}x\sqrt{5}+9 by 126-10\sqrt{5}.

$$factor(\frac{-567x\sqrt{5}+45x\left(\sqrt{5}\right)^{2}+1134-90\sqrt{5}}{15376})$$

The square of \sqrt{5} is 5.

$$factor(\frac{-567x\sqrt{5}+45x\times 5+1134-90\sqrt{5}}{15376})$$

Multiply 45 and 5 to get 225.

$$factor(\frac{-567x\sqrt{5}+225x+1134-90\sqrt{5}}{15376})$$

Consider -567x\times 5^{\frac{1}{2}}+225x+1134-90\times 5^{\frac{1}{2}}. Factor out 9.

$$9\left(-63x\sqrt{5}+25x+126-10\sqrt{5}\right)$$

Rewrite the complete factored expression. Simplify.

$$\frac{9\left(-63x\sqrt{5}+25x+126-10\sqrt{5}\right)}{15376}$$

Evaluate

$$\frac{9\left(63-5\sqrt{5}\right)\left(-\sqrt{5}x+2\right)}{15376}$$