Question

If a² + b² + c² = 24 and ab + bc + ca = – 4, then a + b + c = ________.

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ {a}^2+b^2+c^2=24...\left(1\right) \\ ab+bc+ca=-4...\left(2\right) \\ \mathrm{Now}, \\ {\left(a+b+c\right)}^2 \\ =a^2+b^2+c^2+2\left(ab+bc+ca\right)\left(\mathrm{Using}\mathrm{the}\mathrm{identity}:{\left(a+b+c\right)}^2=a^2+b^2+c^2+2\left(ab+bc+ca\right)\right) \\ =24+2\left(-4\right)\left(\mathrm{From}\left(1\right)\mathrm{and}\left(2\right)\right) \\ =24-8 \\ =16 \\ \mathrm{Since},{\left(a+b+c\right)}^2=16 \\ \mathrm{Therefore},a+b+c=\pm 4. \end{gathered}$$

Hence, if a² + b² + c² = 24 and ab + bc + ca = – 4, then a + b + c = ±4.