What is the minimum value of $2 x^{2} - 8 x + 11$ for real $x$ ?

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The equation $x^{2} - 6 x + k = 0$ has two equal real roots. What is the value of $k$ ?
The graph of $y = x^{2} + k x + 4$ touches the $x$ -axis at exactly one point, and $k > 0$ . What is $k$ ?
The line $y = 2 x + c$ is a tangent to the curve $y = x^{2} - 4 x + 10$ . What is the value of $c$ ?
The roots of $x^{2} - 5 x + 3 = 0$ are $α$ and $β$ . What is the value of $α^{2} + β^{2}$ ?
How many distinct real solutions does $(x^{2} - 4) (x^{2} + 1) (x - 3) = 0$ have?
For which values of $x$ is $x^{2} - 5 x + 6 < 0$ ?

What is the minimum… | TMUA Quadratics and the discriminant