Full Step-by-Step Solutions

1 \(x^2 + 5x + 4 = 0\)
Identify: \(a=1, b=5, c=4\)
\(x = \frac{-5 \pm \sqrt{25 - 16}}{2} = \frac{-5 \pm \sqrt{9}}{2} = \frac{-5 \pm 3}{2}\)
Answers: \(x = -1, x = -4\)

2 \(x^2 - 7x + 10 = 0\)
Identify: \(a=1, b=-7, c=10\)
\(x = \frac{7 \pm \sqrt{49 - 40}}{2} = \frac{7 \pm 3}{2}\)
Answers: \(x = 5, x = 2\)

3 \(x^2 + 2x - 8 = 0\)
Identify: \(a=1, b=2, c=-8\)
\(x = \frac{-2 \pm \sqrt{4 - (-32)}}{2} = \frac{-2 \pm \sqrt{36}}{2} = \frac{-2 \pm 6}{2}\)
Answers: \(x = 2, x = -4\)

4 \(x^2 - 4x - 12 = 0\)
Identify: \(a=1, b=-4, c=-12\)
\(x = \frac{4 \pm \sqrt{16 - (-48)}}{2} = \frac{4 \pm \sqrt{64}}{2} = \frac{4 \pm 8}{2}\)
Answers: \(x = 6, x = -2\)

5 \(2x^2 + 5x + 2 = 0\)
Identify: \(a=2, b=5, c=2\)
\(x = \frac{-5 \pm \sqrt{25 - 16}}{4} = \frac{-5 \pm 3}{4}\)
Answers: \(x = -0.5, x = -2\)

6 \(3x^2 - 10x + 3 = 0\)
Identify: \(a=3, b=-10, c=3\)
\(x = \frac{10 \pm \sqrt{100 - 36}}{6} = \frac{10 \pm 8}{6}\)
Answers: \(x = 3, x = \frac{1}{3} \approx 0.33\)

7 \(x^2 + 6x + 9 = 0\)
Identify: \(a=1, b=6, c=9\)
\(x = \frac{-6 \pm \sqrt{36 - 36}}{2} = \frac{-6 \pm 0}{2}\)
Answers: \(x = -3\) (Repeated root)

8 \(2x^2 - 3x - 5 = 0\)
Identify: \(a=2, b=-3, c=-5\)
\(x = \frac{3 \pm \sqrt{9 - (-40)}}{4} = \frac{3 \pm 7}{4}\)
Answers: \(x = 2.5, x = -1\)

9 \(5x^2 + 13x - 6 = 0\)
Identify: \(a=5, b=13, c=-6\)
\(x = \frac{-13 \pm \sqrt{169 - (-120)}}{10} = \frac{-13 \pm 17}{10}\)
Answers: \(x = 0.4, x = -3\)

10 \(x^2 + 10x + 15 = 0\)
Identify: \(a=1, b=10, c=15\)
\(x = \frac{-10 \pm \sqrt{100 - 60}}{2} = \frac{-10 \pm \sqrt{40}}{2} \approx \frac{-10 \pm 6.32}{2}\)
Answers: \(x \approx -1.84, x \approx -8.16\)

11 \(x^2 + 8x + 12 = 0\)
Rearranged from: \(x^2 + 8x = -12\)
\(x = \frac{-8 \pm \sqrt{64 - 48}}{2} = \frac{-8 \pm 4}{2}\)
Answers: \(x = -2, x = -6\)

12 \(x^2 - 3x - 10 = 0\)
Rearranged from: \(x^2 = 3x + 10\)
\(x = \frac{3 \pm \sqrt{9 - (-40)}}{2} = \frac{3 \pm 7}{2}\)
Answers: \(x = 5, x = -2\)

13 \(2x^2 + 7x - 4 = 0\)
Rearranged from: \(2x^2 + 7x = 4\)
\(x = \frac{-7 \pm \sqrt{49 - (-32)}}{4} = \frac{-7 \pm 9}{4}\)
Answers: \(x = 0.5, x = -4\)

14 \(x^2 - 3x - 10 = 0\)
Rearranged from: \(x^2 - 10 = 3x\)
\(x = \frac{3 \pm \sqrt{9 - (-40)}}{2} = \frac{3 \pm 7}{2}\)
Answers: \(x = 5, x = -2\)

15 \(3x^2 - 2x - 5 = 0\)
Rearranged from: \(3x^2 = 2x + 5\)
\(x = \frac{2 \pm \sqrt{4 - (-60)}}{6} = \frac{2 \pm 8}{6}\)
Answers: \(x = 1.67, x = -1\)

16 \(x^2 + 5x - 14 = 0\)
Expanded from: \(x(x + 5) = 14\)
\(x = \frac{-5 \pm \sqrt{25 - (-56)}}{2} = \frac{-5 \pm 9}{2}\)
Answers: \(x = 2, x = -7\)

17 \(x^2 + 4x - 1 = 0\)
Rearranged from: \(x^2 + 4x = 1\)
\(x = \frac{-4 \pm \sqrt{16 - (-4)}}{2} = \frac{-4 \pm \sqrt{20}}{2} \approx \frac{-4 \pm 4.47}{2}\)
Answers: \(x \approx 0.24, x \approx -4.24\)

18 \(6x^2 - 11x - 7 = 0\)
Rearranged from: \(6x^2 - 7 = 11x\)
\(x = \frac{11 \pm \sqrt{121 - (-168)}}{12} = \frac{11 \pm 17}{12}\)
Answers: \(x = 2.33, x = -0.5\)

19 \(2x^2 - 6x - 8 = 0\)
Expanded from: \(2x(x - 3) = 8\)
\(x = \frac{6 \pm \sqrt{36 - (-64)}}{4} = \frac{6 \pm 10}{4}\)
Answers: \(x = 4, x = -1\)

20 \(4x^2 - 4x + 1 = 0\)
Rearranged from: \(4x^2 + 1 = 4x\)
\(x = \frac{4 \pm \sqrt{16 - 16}}{8} = \frac{4 \pm 0}{8}\)
Answers: \(x = 0.5\) (Repeated root)