1) Solve: x2 + 9x + 20 = 0
x2 + 2x − 3 = 0 ⇒ (x + 4)(x + 5) = 0
- (x + 4) = 0 ⇒ x = −4
- (x + 5) = 0 ⇒ x = −5
x1 = −4 , x2 = −5
2) Solve: x2 + 2x − 3 = 0
x2 + 9x + 20 = 0 ⇒ a = 1, b = 2, c = −3
\color{red}{Δ \ = \ b^2 \ - \ 4ac \ = \ 2^2 \ - \ 4(1)(-3) \ = \ 4 \ - \ (-12) \ = \ 16}
\color{red}{x \ = \ \frac{-b \ ± \ \sqrt{Δ}}{2a} \ = \ \frac{-2 \ ± \ \sqrt{16}}{2} \ ⇒} \color{red}{x_1 \ = \ \frac{-2 \ - \ 4}{2} \ = \ -\frac{6}{2} \ = \ -3 \ , \ }\color{red}{x_2 \ = \ \frac{-2 \ + \ 4}{2} \ = \ \frac{2}{2} \ = \ 1}
3) Solve: 2x^2 \ + \ 7x \ - \ 4 \ = \ 0
\color{red}{x^2 \ + \ 9x \ + \ 20 \ = \ 0 \ ⇒ \ a \ = \ 2, \ b \ = \ 7, \ c \ = \ -4}
\color{red}{Δ \ = \ b^2 \ - \ 4ac \ = \ 7^2 \ - \ 4(2)(-4) \ = \ 49 \ - \ (-32) \ = \ 81}
\color{red}{x \ = \ \frac{-b \ ± \ \sqrt{Δ}}{2a} \ = \ \frac{-7 \ ± \ \sqrt{81}}{4} \ ⇒} \color{red}{x_1 \ = \ \frac{-7 \ - \ 9}{4} \ = \ -\frac{16}{4} \ = \ -4 \ , \ }\color{red}{x_2 \ = \ \frac{-7 \ + \ 9}{4} \ = \ \frac{2}{4} \ = \ \frac{1}{2}}
4) Solve: 2x^2 \ + \ 4x \ + \ 2 \ = \ 0
\color{red}{2x^2 \ + \ 4x \ + \ 2 \ = \ 0 \ ⇒ \ 2x^2 \ + \ 2x \ + \ 2x \ + \ 2 \ = \ 0 \ ⇒ \ 2x(x \ + \ 1) \ + \ 2(x \ + \ 1) \ = \ 0} \color{red}{⇒ \ (x \ + \ 1)(2x \ + \ 2) \ = \ 0}
- \color{red}{(x \ + \ 1) \ = \ 0 \ ⇒ \ x \ = \ -1}
- \color{red}{(2x \ + \ 2) \ = \ 0 \ ⇒ \ x \ = \ -1}
\color{red}{x_1 \ = \ x_2 \ = \ -1}
5) Solve: 18x^2 \ - \ 13x \ + \ 2 \ = \ 0
\color{red}{18x^2 \ - \ 13x \ + \ 2 \ = \ 0 \ ⇒ \ a \ = \ 18, \ b \ = \ -13, \ c \ = \ 2}
\color{red}{Δ \ = \ b^2 \ - \ 4ac \ = \ (-13)^2 \ - \ 4(18)(2) \ = \ 169 \ - \ 144 \ = \ 25}
\color{red}{x \ = \ \frac{-b \ ± \ \sqrt{Δ}}{2a} \ = \ \frac{13 \ ± \ \sqrt{25}}{36} \ ⇒} \color{red}{x_1 \ = \ \frac{13 \ - \ 5}{36} \ = \ \frac{8}{36} \ = \ \frac{2}{9} \ , \ } \color{red}{x_2 \ = \ \frac{13 \ + \ 5}{36} \ = \ \frac{18}{36} \ = \ \frac{1}{2}}
6) Solve: 5x^2 \ + \ 27x \ - \ 18 \ = \ 0
\color{red}{5x^2 \ + \ 27x \ - \ 18 \ = \ 0 \ ⇒ \ a \ = \ 5, \ b \ = \ 27, \ c \ = \ -18}
\color{red}{Δ \ = \ b^2 \ - \ 4ac \ = \ (27)^2 \ - \ 4(5)(-18) \ = \ 729 \ - \ (-360) \ = \ 1089}
\color{red}{x \ = \ \frac{-b \ ± \ \sqrt{Δ}}{2a} \ = \ \frac{-27 \ ± \ \sqrt{1089}}{10} \ ⇒} \color{red}{x_1 \ = \ \frac{-27 \ - \ 33}{10} \ = \ \frac{60}{10} \ = \ 6 \ , \ }\color{red}{x_2 \ = \ \frac{-27 \ + \ 33}{10} \ = \ \frac{6}{10} \ = \ 0.6}
7) Solve: 2x^2 \ + \ 7x \ + \ 3 \ = \ 0
\color{red}{2x^2 \ + \ 7x \ + \ 3 \ = \ 0 \ ⇒ \ 2x^2 \ + \ 6x \ + \ x \ + \ 3 \ = \ 0 \ ⇒ \ 2x(x \ + \ 3) \ + \ 1(x \ + \ 3) \ = \ 0} \color{red}{⇒ \ (2x \ + \ 1)(x \ + \ 3) \ = \ 0}
- \color{red}{(2x \ + \ 1) \ = \ 0 \ ⇒ \ x \ = \ -\frac{1}{2}}
- \color{red}{(x \ + \ 3) \ = \ 0 \ ⇒ \ x \ = \ -3}
\color{red}{x_1 \ = \ -\frac{1}{2} \ , \ x_2 \ = \ -3}
8) Solve: -6x^2 \ + \ 29x \ - \ 35 \ = \ 0
\color{red}{-6x^2 \ + \ 29x \ - \ 35 \ = \ 0 \ ⇒ \ -6x^2 \ + \ 14x \ + \ 15x \ - \ 35 \ = \ 0} \color{red}{ \ ⇒ \ -2x(3x \ - \ 7) \ + \ 5(3x \ - \ 7) \ = \ 0⇒ \ (-2x \ + \ 5)(3x \ - \ 7) \ = \ 0}
- \color{red}{(-2x \ + \ 5) \ = \ 0 \ ⇒ \ x \ = \ \frac{5}{2} \ = \ 2.5}
- \color{red}{(3x \ - \ 7) \ = \ 0 \ ⇒ \ x \ = \ \frac{7}{3}}
\color{red}{x_1 \ = \ 2.5 \ , \ x_2 \ = \ \frac{7}{3}}
9) Solve: 4x^2 \ + \ 8x \ + \ 3 \ = \ 0
\color{red}{4x^2 \ + \ 8x \ + \ 3 \ = \ 0 \ ⇒ \ a \ = \ 4, \ b \ = \ 8, \ c \ = \ 3}
\color{red}{Δ \ = \ b^2 \ - \ 4ac \ = \ (8)^2 \ - \ 4(4)(3) \ = \ 64 \ - \ 48 \ = \ 16}
\color{red}{x \ = \ \frac{-b \ ± \ \sqrt{Δ}}{2a} \ = \ \frac{-8 \ ± \ \sqrt{16}}{8} \ ⇒} \color{red}{x_1 \ = \ \frac{-8 \ - \ 4}{8} \ = \ -\frac{12}{8} \ = \ -\frac{3}{2} \ , \ } \color{red}{x_2 \ = \ \frac{-8 \ + \ 4}{8} \ = \ -\frac{4}{8} \ = \ -\frac{1}{2}}
10) Solve: x^2 \ + \ 3x \ - \ 4 \ = \ 0
\color{red}{x^2 \ + \ 3x \ - \ 4 \ = \ 0 \ ⇒ \ a \ = \ 1, \ b \ = \ 3, \ c \ = \ -4}
\color{red}{Δ \ = \ b^2 \ - \ 4ac \ = \ (3)^2 \ - \ 4(1)(-4) \ = \ 9 \ + \ 16 \ = \ 25}
\color{red}{x \ = \ \frac{-b \ ± \ \sqrt{Δ}}{2a} \ = \ \frac{-3 \ ± \ \sqrt{25}}{2} \ ⇒} \color{red}{x_1 \ = \ \frac{-3 \ - \ 5}{2} \ = \ -\frac{9}{2} \ = \ -4.5 \ , \ } \color{red}{x_2 \ = \ \frac{-3 \ + \ 5}{2} \ = \ \frac{2}{2} \ = \ 1}