Answer:
A. By extracting square root
1.5x²- 45 = 0
[tex] \frac{5x²}{5} = \frac{45}{5} \\ \sqrt{{x}^{2}} = \sqrt{ 9} \\ \boxed{ x = 3,-3}[/tex]
[tex] \red{⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉}[/tex]
2.(m - 2)² = 16
[tex]\sqrt{ (m - 2)² }= \sqrt{16} \\ m - 2 = ±4 \\m = 2±4\\ \boxed{ m = 6, - 2}[/tex]
[tex]\red{⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉}[/tex]
B. by factoring
1.11p + 15 = -2p²
[tex]2p² + 11p + 15 = 0 \\ 2p {}^{2} + 6p + 5p + 15 \\ 2p(p + 3) + 5(p + 3) \\ (2p + 5)(p + 3) = 0 \\ \boxed{p = - \frac{5}{2}, - 3}[/tex]
[tex]\purple{⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉}[/tex]
2.2k² - 14 = -3k
[tex]2k² +3k - 14 =0 \\ 2k² + 7k - 4k- 14 =0 \\ k(2k + 7) - 2(2k + 7) = 0 \\ (2k + 7)(k - 2) = 0 \\ \boxed{ k = - \frac{7}{2} ,2}[/tex]
[tex]\purple{⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉}[/tex]
C. by completing the square
1.x² + 15x + 15 = 2 + x
[tex]x² + 15x - x + 15 - 2 = 0 \\ {x}^{2} + 14x + 13 = 0 \\ {x}^{2} + 14x = - 13\\ ( \frac{b}{2} {)}^{2} = (\frac{14}{2} ) {}^{2} = (7) {}^{2} = 49 \\ {x}^{2} + 14x + 49 = - 13 + 49 \\ (x + 7) {}^{2} = 36 \\ \sqrt{(x + 7) {}^{2} } = \sqrt{36} \\ x + 7 = 6 \\ x = - 7±6 \\ \boxed{ x = - 1, - 13}[/tex]
[tex]\green{⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉}[/tex]
2.-3n² + 4n -59 = -4n²
[tex]4n²-3n² + 4n -59 =0 \\ n² + 4n - 59 = 0 \\ n² + 4n = 59 \\ ( \frac{b}{2} ) {}^{2} = (\frac{4}{2} ) {}^{2} = (2 {)}^{2} = 4 \\ n² + 4n + 4 = 59 + 4 \\ (n + 2) {}^{2} = 63 \\ \sqrt{ (n + 2) {}^{2} } = \sqrt{63} \\ n + 2 = ±3 \sqrt{7} \\ \boxed{ n = - 2±3 \sqrt{7} }[/tex]
[tex]\green{⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉}[/tex]
D.by using the quadratic formula
1.3x² - 5x - 8 = 0
[tex]3x² - 5x - 8 = 0 \\ \small{x = \frac{ - ( - 5)± \sqrt{( - 5) {}^{2} - 4(3)( - 8)} }{2(3)} } \\ x = \frac{5± \sqrt{25 + 96} }{6} \\ x = \frac{5± \sqrt{121} }{6} \\ x = \frac{5±11}{6} \\ \boxed{x = \frac{8}{3} , - 1}[/tex]
[tex]\orange{⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉}[/tex]
2.11x² + 4x - 52 = 10x² - 7
[tex]11x² -10x² + 4x - 52 + 7= 0 \\ {x}^{2} + 4x - 45 = 0 \\ x = \frac{ - 4± \sqrt{(4) {}^{2} - 4(1)( - 45)} }{2(1)} \\ x = \frac{ - 4± \sqrt{16 + 180} }{2} \\ x = \frac{ - 4± \sqrt{196} }{2} \\ x = \frac{ - 4±14}{2} \\ \boxed{ x = 5, - 9}[/tex]
[tex]\orange{⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉⑉}[/tex]
