Let $a,b,c$ are real numbers with $a^{2} + b^{2} + c^{2} 0$. Then the equation $x^{2} + (a + b

Question

Let $a,b,c$ are real numbers with $a^{2} + b^{2} + c^{2} > 0$. Then the equation $x^{2} + (a + b + c)x + \left( a^{2} + b^{2} + c^{2} \right) = 0$ has

Accepted Answer

$D = (a + b + c)^{2} - 4\left( a^{2} + b^{2} + c^{2} \right) = - 3\left( a^{2} + b^{2} + c^{2} \right) + 2(ab + bc + ca)$

$$\begin{matrix} & \ = - \left( a^{2} + b^{2} + c^{2} \right) - 2\left( a^{2} + b^{2} + c^{2} - ab - bc - ca \right) \\ & \ = - \left( a^{2} + b^{2} + c^{2} \right) - \left( (a - b)^{2} + (b - c)^{2} + (c - a)^{2} \right) \\ \Rightarrow \ D & \ < 0 \end{matrix}$$

Question

Options

Solution

More Quadratic Equation Questions