Formulele de calcul prescurtat sunt niște tehnici care oferă posibilitatea unui calcul mai rapid al expresiilor care conțin radicali sau permit scrierea radicalilor dubli sub forma unor expresii ce conțin radicali simpli.
Formulele de calcul sunt următoarele:
- \(\left( a+b \right)^{2}=a^{2}+2ab+b^{2}\)
Demonstrație
\(\left( a+b \right)^{2}=\left( a+b \right)\left( a+b \right)=\)
\(a^{2}+ab+ba+b^{2}=a^{2}+2ab+b^{2}\)
Exemple:
\(\left( 2+\sqrt{3} \right)^{2}\)
\(\text{avem }a=2,b=\sqrt{3}\)
\(\left( 2+\sqrt{3} \right)^{2}=2^{2}+2\cdot 2\cdot \sqrt{3}+\sqrt{3}^{2}=\)
\(=4+4\sqrt{3}+3=7+4\sqrt{3}\)
\(\left( \sqrt{3}+\sqrt{5} \right)^{2}\)
\(\text{avem }a=\sqrt{3},b=\sqrt{5}\)
\(\left( \sqrt{3}+\sqrt{5} \right)^{2}=\sqrt{3}^{2}+2\cdot \sqrt{3}\cdot \sqrt{5}+\sqrt{5}^{2}=\)
\(=3+2\sqrt{15}+5=8+2\sqrt{15}\)
\(\left( 2\sqrt{5}+3\sqrt{2} \right)^{2}\)
\(\text{avem }a=2\sqrt{5},b=3\sqrt{2}\)
\(\left( 2\sqrt{5}+3\sqrt{2} \right)^{2}=\left( 2\sqrt{5} \right)^{2}+2\cdot 2\sqrt{5}\cdot 3\sqrt{2}+\left( 3\sqrt{2} \right)^{2}=\)
\(=2^{2}\cdot \sqrt{5}^{2}+\left( 2\cdot 2\cdot 3 \right)\sqrt{5\cdot 2}+3^{2}\cdot \sqrt{2}^{2}=\)
\(=4\cdot 5+12\sqrt{10}+9\cdot 2=\)
\(=20+12\sqrt{10}+18=38+12\sqrt{10}\)
Formulă de calcul:
- \(\left( a-b \right)^{2}=a^{2}-2ab+b^{2}\)
Demonstrație
\(\left( a-b \right)^{2}=\left( a-b \right)\left( a-b \right)=\)
\(=a^{2}-ab-ba+b^{2}=a^{2}-2ab+b^{2}\)
Exemple:
\(\left( 3-\sqrt{5} \right)^{2}\)
\(\text{avem }a=3,b=\sqrt{5}\)
\(\left( 3-\sqrt{5} \right)^{2}=3^{2}-2\cdot 3\cdot \sqrt{5}+\sqrt{5}^{2}=\)
\(=9-6\sqrt{5}+5=13-6\sqrt{5}\)
\(\left( \sqrt{2}-\sqrt{7} \right)^{2}\)
\(\text{avem }a=\sqrt{2},b=\sqrt{7}\)
\(\left( \sqrt{2}-\sqrt{7} \right)^{2}=\sqrt{2}^{2}-2\cdot \sqrt{2}\cdot \sqrt{7}+\sqrt{7}^{2}=\)
\(=2-2\sqrt{14}+7=9-2\sqrt{14}\)
\(\left( 3\sqrt{7}-2\sqrt{5} \right)^{2}\)
\(\text{avem }a=3\sqrt{7},b=2\sqrt{5}\)
\(\left( 3\sqrt{7}-2\sqrt{5} \right)^{2}=\left( 3\sqrt{7} \right)^{2}-2\cdot 3\sqrt{7}\cdot 2\sqrt{5}+\left( 2\sqrt{5} \right)^{2}=\)
\(=3^{2}\cdot \sqrt{7}^{2}-\left( 2\cdot 3\cdot 2\right)\sqrt{7\cdot 5}+2^{2}\cdot \sqrt{5}^{2}=\)
\(=9\cdot 7-12\sqrt{35}+4\cdot 5=63-12\sqrt{35}+20=\)
\(=83-12\sqrt{35}\)
Formulă de calcul:
- \(\left( a+b \right)\left( a-b \right)=a^{2}-b^{2}\)
Demonstrație
\(\left( a+b \right)\left( a-b \right)=a^{2}-ab+ab-b^{2}=a^{2}-b^{2}\)
Exemple:
\(\left( \sqrt{5}+\sqrt{3} \right)\left( \sqrt{5}-\sqrt{3} \right)\)
\(\text{avem }a=\sqrt{5},b=\sqrt{3}\)
\(\left( \sqrt{5}+\sqrt{3} \right)\left( \sqrt{5}-\sqrt{3} \right)=\sqrt{5}^{2}-\sqrt{3}^{2}=\)
\(=5-3=2\)
\(\left( 2\sqrt{3}+4\sqrt{7} \right)\left( 2\sqrt{3}-4\sqrt{7} \right)\)
\(\text{avem }a=2\sqrt{3},b=4\sqrt{7}\)
\(\left( 2\sqrt{3}+4\sqrt{7} \right)\left( 2\sqrt{3}-4\sqrt{7} \right)=2^{2}\sqrt{3}^{2}-4^{2}\sqrt{7}^{2}=\)
\(=4\cdot 3-16\cdot 7=12-112=-100\)
Formulă de calcul:
- \(\sqrt{A\pm \sqrt{B}}=\sqrt{\frac{A+C}{2}}\pm \sqrt{\frac{A-C}{2}},\text{ unde }C=\sqrt{A^{2}-B}\)
Exemple:
\(\sqrt{7\ +2\sqrt{6}}=\)
\(\text{avem }A=7,B=2\sqrt{6}=\sqrt{24}\)
\(\Rightarrow C=\sqrt{7^{2}-24}=\sqrt{49-24}=\sqrt{25}=5\)
\(\sqrt{7\ +2\sqrt{6}}=\sqrt{\frac{7+5}{2}}+ \sqrt{\frac{7-5}{2}}=\)
\(=\sqrt{\frac{12}{2}}+\sqrt{\frac{2}{2}}=\sqrt{6}+1\)
\(\sqrt{14\ -6\sqrt{5}}=\)
\(\text{avem }A=14,B=6\sqrt{5}=\sqrt{180}\)
\(\Rightarrow C=\sqrt{14^{2}-180}=\sqrt{196-180}=\sqrt{16}=4\)
\(\sqrt{14\ -6\sqrt{5}}=\sqrt{\frac{196+4}{2}}- \sqrt{\frac{196-4}{2}}=\)
\(=\sqrt{\frac{200}{2}}-\sqrt{\frac{192}{2}}=\sqrt{100}-\sqrt{96}=10-4\sqrt{6}\)