Method of Completing Square

To find the roots of a quadratic equation by completing the perfect square, the first condition is the coefficient of $x^2$, that is $a$ must be equal to 1 to make it easy to factor. If $a$ is not 1, divide the equation by the coefficient $x^2$ so that $a$ equals 1.

If the term on the left side of the equation appears  $x^2+2(\frac{b}{2})x+(\frac{b}{2})^2$, then factor it becomes $(x+\frac{b}{2})^2$.

Which this factoring is valid because $x^2+2(\frac{b}{2})x+(\frac{b}{2})^2=(x+\frac{b}{2})^2$.

Furthermore, the steps to complete a perfect square are as follows:

-On quadratic equations (with $a=1$) $x^2+bx+c=0$, subtract left and right sides by $c$ (if $c <0$, this will be increments)

$x^2+bx+c-c=0-c$

$x^2+bx=-c$

-Modify the term of $bx$ to $2(\frac{b}{2})x$

$x^2+2(\frac{b}{2})x = –c$

-Add the term of $(\frac{b}{2})^2$ on the left and right side

$x^2+2(\frac{b}{2})x+(\frac{b}{2})^2  =(\frac{b}{2})^2–c$

- Change the left side of the equation to $(x+\frac{b}{2})^2$.

$(x+\frac{b}{2})^2=(\frac{b}{2})^2–c$

- Take the roots of the left and right segments

 $x+\frac{b}{2}=±√((\frac{b}{2})^2–c)$

- Subtract left and right sides by $\frac{b}{2}$ (if $b <0$, this will be increments)

The ± sign means that the formula handles two operations, namely addition, and subtraction.

The root $𝑥₁$ is obtained by taking the plus sign (+):

-Root $𝑥₂$ is obtained by taking the minus sign (-) :

Example 1. Find the roots of the equations

$x^2-5x+4=0$

$x^2+7x+12=0$

Answer:

Example 2. Find the roots of the equations

$2x^2-7x+3=0$

$6x^2-5x-6=0$

Answer: