Solve the equation:
[tex]4x^2+5x+2=2x^2+7x-1[/tex]We need to take all terms of the equation to the left side. We do that by subtracting each one of them from both sides of the equation as follows:
[tex]4x^2+5x+2-2x^2-7x+1=0[/tex]Collecting like terms:
[tex]2x^2-2x+3=0[/tex]Identify the coefficients: a = 2, b = -2, c = 3. And apply the quadratic formula:
[tex]$x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}$[/tex]Substituting:
[tex]x=\frac{-(-2)\pm\sqrt{(-2)^2-4(2)(3)}}{2(2)}[/tex]Operating:
[tex]\begin{gathered} x=\frac{2\pm\sqrt{4-24}}{4} \\ \\ x=\frac{2\pm\sqrt{-20}}{4} \end{gathered}[/tex]The solutions to the equation are complex because the square root of a negative number is imaginary:
[tex]x=\frac{2\pm\sqrt{20}i}{4}[/tex]Since 20 = 4*5, this expression can be simplified:
[tex]\begin{gathered} x=\frac{2\pm2\sqrt{5}i}{4} \\ \\ x=\frac{1\pm\sqrt{5}\imaginaryI}{2} \end{gathered}[/tex]Finally, separating the real and imaginary parts:
[tex]x=\frac{1}{2}\pm\frac{\sqrt{5}}{2}i[/tex]