Question

What are the steps to solve the quadratic equation x^2+4x-6=0

Answer 1

`METHOD\ 1:\\\\Use:\\\\(*)\qquad(a+b)^2=a^2+2ab+b^2\\---------------------\\\\x^2+4x-6=0\qquad\text{add 6 to both sides}\\\\x^2+4x=6\\\\x^2+2(x)(2)=6\qquad\text{add}\ 2^2\ \text{to both sides}\\\\\underbrace{x^2+2(x)(2)+2^2}_((*))=6+2^2\\\\(x+2)^2=6+4\\\\(x+2)^2=10\Rightarrow x+2=\pm√(10)\qquad\text{subtract 2 from both sides}\\\\\boxed{x=-2-√(10)\ \vee\ x=-2+√(10)}`

`METHOD\ 2:\\\\\text{use the quadratic formula:}\\\\ax^2+bx+c=0\\\\\Delta=b^2-4ac\\\\if\ \Delta>0,\ \text{the equation has two solutions}\ x=(-b\pm\sqrt\Delta)/(2a)\\\\if\ \Delta=0,\ \text{then the equation has one solution:}\ x=(-b)/(2a)\\\\if\ \Delta<0,\ \text{then the equation has not solution}`

`x^2+4x-6=0\\\\a=1,\ b=4,\ c=-6\\\\\Delta=4^2-4(1)(-6)=16+24=40>0\\\\\sqrt\Delta=√(40)=√(4\cdot10)=\sqrt4\cdot√(10)=2√(10)\\\\x_1=(-4-2√(10))/(2(1))=(-4)/(2)-(2√(10))/(2)=-2-√(10)\\\\x_2=(-4+2√(10))/(2(1))=(-4)/(2)+(2√(10))/(2)=-2+√(10)`