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Solve a quadratic inequality
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$x ^ { 2 } + x + 1 > 0$
$x ^ { 2 } + x + 1 > 0$
Solution of inequality
$x \in \mathbb{R} \left ( \text{It holds for all real numbers} \right )$
$x \in \mathbb{R} \left ( \text{It holds for all real numbers} \right )$
$ $ Solve a solution to $ x$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } > \color{#FF6800}{ 0 }$
$ $ Convert the inequality to a quadratic equation to find $ x_{1}, x_{2}$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } = \color{#FF6800}{ 0 }$
$ $ Calculate using the quadratic formula $ $
$x \in \emptyset \left ( \text{Do not have the solution} \right )$
$\color{#FF6800}{ x } \in \emptyset \left ( \text{Do not have the solution} \right )$
$ $ If there is no real root, the left side of the inequality has always a positive value or a negative value depending on coefficient of the leading highest term a(= $ 1 $ ) of $ ax^{2}+bx+c=0$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } > \color{#FF6800}{ 0 }$
$\color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 1 } > \color{#FF6800}{ 0 }$
$ $ Since the coefficient of the leading highest term (= $ 1 $ ) is $ $ (+) $ $ , the left side of the inequality is always $ $ (+) $ $ So this inequality is $ $ TRUE $ $ regardless of the value of $ x$
$\color{#FF6800}{ x } \in \mathbb{R} \left ( \text{It holds for all real numbers} \right )$
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