Question

Solve x^2+1/2x+1/16=4/9

Answer 1

`\large\boxed{x=-(11)/(12)\ \vee\ x=(5)/(12)}`

Explanation:

`x^2+(1)/(2)x+(1)/(16)=(4)/(9)\\\\x^2+2(x)\left((1)/(4)\right)+\left((1)/(4)\right)^2=(4)/(9)\qquad\text{use}\ (a+b)^2=a^2+2ab+b^2\\\\\left(x+(1)/(4)\right)^2=(4)/(9)\iff x+(1)/(4)=\pm\sqrt{(4)/(9)}\\\\x+(1)/(4)=-(2)/(3)\ \vee\ x+(1)/(4)=(2)/(3)\qquad\text{multiply each sides  by}\ (3)(4)=12`

`12x+12\!\!\!\!\!\diagup^3\cdot(1)/(4\!\!\!\!\diagup_1)=12\!\!\!\!\!\diagup^4\cdot\left(-(2)/(3\!\!\!\!\diagup_1)\right)\ \vee\ 12x+12\!\!\!\!\!\diagup^3\cdot(1)/(4\!\!\!\!\diagup_1)=12\!\!\!\!\!\diagup^4\cdot\left((2)/(3\!\!\!\!\diagup_1)\right)\\\\12x+3=-8\ \vee\ 12x+3=8\qquad\text{subtract 3 from each sides}\\\\12x=-11\ \vee\ 12x=5\qquad\text{divide each sides by 12}\\\\x=-(11)/(12)\ \vee\ x=(5)/(12)`