Answer:
The first step is to divide all the terms by the coefficient of [tex]x^{2}[/tex] which is 2.
The solutions to the quadratic equation [tex]2x^2\:-\:5x\:+\:67\:=\:0[/tex] are:
[tex]x=\frac{5}{4}+i\frac{\sqrt{511}}{4},\:x=\frac{5}{4}-i\frac{\sqrt{511}}{4}[/tex]
Step-by-step explanation:
Considering the equation
[tex]2x^2\:-\:5x\:+\:67\:=\:0[/tex]
The first step is to divide all the terms by the coefficient of [tex]x^{2}[/tex] which is 2.
so
[tex]\frac{2x^2-5x}{2}=\frac{-67}{2}[/tex]
[tex]x^2-\frac{5x}{2}=-\frac{67}{2}[/tex]
Lets now solve the equation by completeing the remaining steps
Write equation in the form: [tex]x^2+2ax+a^2=\left(x+a\right)^2[/tex]
Solving for [tex]a[/tex],
[tex]2ax=-\frac{5}{2}x[/tex]
[tex]a=-\frac{5}{4}[/tex]
[tex]\mathrm{Add\:}a^2=\left(-\frac{5}{4}\right)^2\mathrm{\:to\:both\:sides}[/tex]
[tex]x^2-\frac{5x}{2}+\left(-\frac{5}{4}\right)^2=-\frac{67}{2}+\left(-\frac{5}{4}\right)^2[/tex]
[tex]x^2-\frac{5x}{2}+\left(-\frac{5}{4}\right)^2=-\frac{511}{16}[/tex]
Completing the square
[tex]\left(x-\frac{5}{4}\right)^2=-\frac{511}{16}[/tex]
Since, you had required to know the first step in completing the square for the equation above, I hope you have got the point, but let me quickly solve the remaining solution.
For [tex]f^2\left(x\right)=a[/tex] the solution are [tex]f\left(x\right)=\sqrt{a},\:-\sqrt{a}[/tex]
Solving
[tex]x-\frac{5}{4}=\sqrt{-\frac{511}{16}}[/tex]
[tex]x-\frac{5}{4}=\sqrt{-1}\sqrt{\frac{511}{16}}[/tex]
[tex]x-\frac{5}{4}=i\sqrt{\frac{511}{16}}[/tex] ∵ Applying imaginary number rule [tex]\sqrt{-1}=i[/tex]
[tex]x-\frac{5}{4}=i\frac{\sqrt{511}}{\sqrt{16}}[/tex]
[tex]-\frac{5}{4}=i\frac{\sqrt{511}}{4}[/tex]
[tex]x=\frac{5}{4}+i\frac{\sqrt{511}}{4}[/tex]
Similarly, solving
[tex]x-\frac{5}{4}=-\sqrt{-\frac{511}{16}}[/tex]
[tex]x-\frac{5}{4}=-i\frac{\sqrt{511}}{4}[/tex] ∵ Applying imaginary number rule [tex]\sqrt{-1}=i[/tex]
[tex]x=\frac{5}{4}-i\frac{\sqrt{511}}{4}[/tex]
Therefore, the solutions to the quadratic equation are:
[tex]x=\frac{5}{4}+i\frac{\sqrt{511}}{4},\:x=\frac{5}{4}-i\frac{\sqrt{511}}{4}[/tex]
