Explicație pas cu pas:
1)
[tex]4{x}^{2} + 2x - 2 > 0 \\ 2(2 {x}^{2} + x - 1) > 0 \\ 2(x + 1)(2x - 1) > 0 \\ 2(x + 1)(2x - 1) = 0 [/tex]
[tex]x_{1} = - 1 \\ x_{2} = - \frac{1}{2} \\ [/tex]
[tex]- \infty < x < - 1 \: sau \: \frac{1}{2} < x < + \infty \\[/tex]
2)
[tex] - 2 {x}^{2} + 3x + 3 < 0[/tex]
[tex]- 2 {x}^{2} + 3x + 3 = 0[/tex]
Δ = 3² + 4•2•3 = 9 + 24 = 33
[tex]x_{1} = \frac{ - 3 + \sqrt{33} }{- 4} = \frac{3 - \sqrt{33} }{4} \\x_{2} = \frac{ - 3 - \sqrt{33} }{- 4} = \frac{3 + \sqrt{33} }{4} [/tex]
[tex]- \infty < x < \frac{3 - \sqrt{33} }{4} \: sau \: \frac{3 + \sqrt{33} }{4} < x < + \infty \\[/tex]
3)
[tex]- {x}^{2} + 4x - 9 \geqslant 0 \\ - (x - 2)^{2} - 5 \geqslant 0[/tex]
[tex]fara \: solutii \: in \: multimea \: numerelor \: reale[/tex]
4)
[tex]4{x}^{2} - 5x + 12 \leqslant 0[/tex]
[tex]4{x}^{2} - 5x + 12 = 0[/tex]
Δ = (-5)² -4•4•12 = 25 - 192 = -167 < 0
fără soluții în mulțimea numerelor reale
5)
[tex]- 2{x}^{2} - 3x + 1 \geqslant 0[/tex]
[tex]- 2{x}^{2} - 3x + 1 = 0[/tex]
Δ = (-3)² + 4•2 = 9 + 8 = 17
[tex]x_{1} = \frac{3 + \sqrt{17} }{ - 4} = - \frac{3 + \sqrt{17} }{4} \\ x_{1} = \frac{3 - \sqrt{17} }{ - 4} = - \frac{ 3 - \sqrt{17}}{4} [/tex]
[tex]- \frac{3 - \sqrt{17}}{4} \leqslant x \leqslant - \frac{3 + \sqrt{17}}{4} \\[/tex]
6)
[tex]{x}^{2} + x + 1 > 0[/tex]
[tex]{\left(x + \frac{1}{2} \right)}^{2} + \frac{3}{4} > 0 \\ [/tex]
[tex]- \infty < x < + \infty[/tex]
7)
[tex]- {x}^{2} + x - 1 < 0 \\ - {\left(x - \frac{1}{2} \right)}^{2} - \frac{3}{4} < 0[/tex]
[tex] - \infty < x < + \infty [/tex]
8)
[tex]{x}^{2} - x - 1 \leqslant 0 [/tex]
[tex]{x}^{2} - x - 1 = 0[/tex]
Δ = 1 + 4 = 5
[tex]x_{1} = \frac{1 - \sqrt{5} }{2} \\ x_{1} = \frac{1 + \sqrt{5} }{2} [/tex]
[tex]\frac{1 - \sqrt{5} }{2} \leqslant x \leqslant \frac{1 + \sqrt{5} }{2} \\ [/tex]
9)
[tex]{x}^{2} - x + 1 \geqslant 0[/tex]
[tex]{\left(x - \frac{1}{2} \right)}^{2} + \frac{3}{4} \geqslant 0 \\ [/tex]
[tex] - \infty < x < + \infty [/tex]
10)
[tex]-{x}^{2} - x - 1 \geqslant 0[/tex]
[tex]- {\left(x + \frac{1}{2} \right)}^{2} - \frac{3}{4} \geqslant 0 \\ [/tex]
fără soluții în mulțimea numerelor reale
