Which are the solutions of x^2 = –5x + 8?
Answer: The required solution is
\(x=\dfrac{-5\pm\sqrt{57}}{2}.\)
Step-by-step explanation: We are given to solve the following differential equation :
\(x^2=-5x+8~~~~~~~~~~~~~~~~~~~~~~~(i)\)
We will be using the method of completing the square to solve the given equation.
From equation (i), we have
\(x^2=-5x+8\\\\\Rightarrow x^2+5x=8\\\\\Rightarrow x^2+2\times x\times \dfrac{5}{2}+\left(\dfrac{5}{2}\right)^2=8+\left(\dfrac{5}{2}\right)^2\\\\\\\Rightarrow \left(x+\dfrac{5}{2}\right)^2=8+\dfrac{25}{4}\\\\\\\Rightarrow \left(x+\dfrac{5}{2}\right)^2=\dfrac{57}{4}\\\\\\\Rightarrow x+\dfrac{5}{2}=\pm\dfrac{\sqrt{57}}{2}\\\\\\\Rightarrow x=-\dfrac{5}{2}\pm\dfrac{\sqrt{57}}{2}\\\\\\\Rightarrow x=\dfrac{-5\pm\sqrt{57}}{2}.\)
Thus, the required solution is
\(x=\dfrac{-5\pm\sqrt{57}}{2}.\)