Issolvi {left(sqrt{2}+sqrt{5}right)}^2-{left(2+sqrt{10}right)}^2+sqrt{90}+left(2sqrt{2}-1right)left(2sqrt{2}+1right)

$${ \left( \sqrt{ 2 } + \sqrt{ 5 } \right) }^{ 2 } - { \left(2+ \sqrt{ 10 } \right) }^{ 2 } + \sqrt{ 90 } + \left( 2 \sqrt{ 2 } -1 \right) \left( 2 \sqrt{ 2 } +1 \right)$$

Evalwa

$$\sqrt{10}\approx 3.16227766$$

Passi tas-Soluzzjonijiet

Uża teorema binomjali \left(a+b\right)^{2}=a^{2}+2ab+b^{2} biex tespandi \left(\sqrt{2}+\sqrt{5}\right)^{2}.

$$\left(\sqrt{2}\right)^{2}+2\sqrt{2}\sqrt{5}+\left(\sqrt{5}\right)^{2}-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Il-kwadrat ta' \sqrt{2} huwa 2.

$$2+2\sqrt{2}\sqrt{5}+\left(\sqrt{5}\right)^{2}-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Biex timmultiplika \sqrt{2} u \sqrt{5}, immultiplika n-numri taħt l-għerq kwadrat.

$$2+2\sqrt{10}+\left(\sqrt{5}\right)^{2}-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Il-kwadrat ta' \sqrt{5} huwa 5.

$$2+2\sqrt{10}+5-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Żid 2 u 5 biex tikseb 7.

$$7+2\sqrt{10}-\left(2+\sqrt{10}\right)^{2}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Uża teorema binomjali \left(a+b\right)^{2}=a^{2}+2ab+b^{2} biex tespandi \left(2+\sqrt{10}\right)^{2}.

$$7+2\sqrt{10}-\left(4+4\sqrt{10}+\left(\sqrt{10}\right)^{2}\right)+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Il-kwadrat ta' \sqrt{10} huwa 10.

$$7+2\sqrt{10}-\left(4+4\sqrt{10}+10\right)+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Żid 4 u 10 biex tikseb 14.

$$7+2\sqrt{10}-\left(14+4\sqrt{10}\right)+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Biex issib l-oppost ta' 14+4\sqrt{10}, sib l-oppost ta' kull terminu.

$$7+2\sqrt{10}-14-4\sqrt{10}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Naqqas 14 minn 7 biex tikseb -7.

$$-7+2\sqrt{10}-4\sqrt{10}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Ikkombina 2\sqrt{10} u -4\sqrt{10} biex tikseb -2\sqrt{10}.

$$-7-2\sqrt{10}+\sqrt{90}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Iffattura 90=3^{2}\times 10. Erġa' ikteb l-għerq kwadrat tal-prodott \sqrt{3^{2}\times 10} bħala l-prodott tal-għeruq kwadrati \sqrt{3^{2}}\sqrt{10}. Ħu l-għerq kwadrat ta' 3^{2}.

$$-7-2\sqrt{10}+3\sqrt{10}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Ikkombina -2\sqrt{10} u 3\sqrt{10} biex tikseb \sqrt{10}.

$$-7+\sqrt{10}+\left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right)$$

Ikkunsidra li \left(2\sqrt{2}-1\right)\left(2\sqrt{2}+1\right). Il-multiplikazzjoni tista' tiġi ttrasformata fid-differenza tal-kwadrati li jużaw ir-regola: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Ikkwadra 1.

$$-7+\sqrt{10}+\left(2\sqrt{2}\right)^{2}-1$$

Espandi \left(2\sqrt{2}\right)^{2}.

$$-7+\sqrt{10}+2^{2}\left(\sqrt{2}\right)^{2}-1$$

Ikkalkula 2 bil-power ta' 2 u tikseb 4.

$$-7+\sqrt{10}+4\left(\sqrt{2}\right)^{2}-1$$

Il-kwadrat ta' \sqrt{2} huwa 2.

$$-7+\sqrt{10}+4\times 2-1$$

Immultiplika 4 u 2 biex tikseb 8.

$$-7+\sqrt{10}+8-1$$

Naqqas 1 minn 8 biex tikseb 7.

$$-7+\sqrt{10}+7$$

Żid -7 u 7 biex tikseb 0.

$$\sqrt{10}$$