1.
(a - b)² = a² - 2ab + b²
(a + b)² = a² + 2ab + b²
(a + b)(a - b) = a² - b²
[tex]a) \ (x-8)^{2} = x^{2}-2\cdot x \cdot8 + 8^{2} = x^{2}-16x+64\\\\b) \ (2x+6y)^{2} = (2x)^{2}+2\cdot2x\cdot6y + (6y)^{2} = 4x^{2}-24xy + 36y^{2}\\\\c) \ (x+8)(x-8) = x^{2}-8^{2} = x^{2}-64[/tex]
2.
[tex]a) \ (2\sqrt{2}-6)^{2} = (2\sqrt{2})^{2}-2\cdot2\sqrt{2}\cdot6+6^{2} = 8-24\sqrt{2}+36 = 44-24\sqrt{2}\\\\b) \ |1|-|\frac{3}{5}|+|-3| = 1 - \frac{3}{5}-(-3) = 1-\frac{3}{5}+3 = 4-\frac{3}{5} = 3\frac{2}{5}\\\\c) \ |\sqrt{7}-1| = \sqrt{7}-1[/tex]
3.
[tex]\frac{2}{6+\sqrt{3}} = \frac{2}{6+\sqrt{3}}\cdot\frac{6-\sqrt{3}}{6-\sqrt{3}} = \frac{2(6-\sqrt{3})}{36-3} = \frac{12-2\sqrt{3}}{33}[/tex]
4.
a)
[tex](3x+1)^{2}+(2x-4)^{2} = 3-13x^{2}\\\\9x^{2}+6x+1+4x^{2}-16x+16 = 3-13x^{2}\\\\13x^{2}-10x+17=3-13x^{2}\\\\13x^{2}+13x^{2}-10x+17-3=0\\\\26x^{2}-10x+14 = 0 \ \ /:2\\\\13x^{2}-5x+7 = 0\\\\\Delta = b^{2}-4ac = (-5)^{2}-4\cdot13\cdot7 = 25 - 364 = -339 < 0, \ brak \ pierwiastow[/tex]
b)
[tex]4|x| -1 = 5\\\\4|x| = 5+1\\\\4|x| = 6 \ \ /:4\\\\|x| = 1,5\\\\x = 1,5 \ \vee \ x = -1,5[/tex]
5.
[tex]|x| > 8\\\\x > 8 \ \ \vee \ \ -x>8 \ \ /:(-1)\\\\x > 8 \ \ \vee \ \ x <-8\\\\x\in (-\infty; -8) \ \cup \ (8; +\infty)[/tex]
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