how to complete the square

To complete the square for a quadratic expression or equation, follow these steps:

1. Start with the quadratic expression in standard form:

ax2+bx+cax^2+bx+cax2+bx+c

If the coefficient aaa is not 1, divide the entire expression by aaa to make the coefficient of x2x^2x2 equal to 1.

2. Isolate the constant term:
   Rewrite the equation so that the constant term ccc is on the right side:

x2+bax=−cax^2+\frac{b}{a}x=-\frac{c}{a}x2+ab​x=−ac​

3. Find half of the coefficient of xxx:
   Take the coefficient of xxx, which is ba\frac{b}{a}ab​, divide it by 2:

b2a\frac{b}{2a}2ab​

4. Square this half:

(b2a)2\left(\frac{b}{2a}\right)^2(2ab​)2

5. Add and subtract this square inside the equation:
   Add (b2a)2\left(\frac{b}{2a}\right)^2(2ab​)2 to both sides to keep the equation balanced:

x2+bax+(b2a)2=−ca+(b2a)2x^2+\frac{b}{a}x+\left(\frac{b}{2a}\right)^2=-\frac{c}{a}+\left(\frac{b}{2a}\right)^2x2+ab​x+(2ab​)2=−ac​+(2ab​)2

6. Rewrite the left side as a perfect square:
   The left side factors into:

(x+b2a)2\left(x+\frac{b}{2a}\right)^2(x+2ab​)2

7. Solve for xxx by taking the square root of both sides:

x+b2a=±−ca+(b2a)2x+\frac{b}{2a}=\pm \sqrt{-\frac{c}{a}+\left(\frac{b}{2a}\right)^2}x+2ab​=±−ac​+(2ab​)2​

8. Isolate xxx:

x=−b2a±b2−4ac4a2=−b±b2−4ac2ax=-\frac{b}{2a}\pm \sqrt{\frac{b^2-4ac}{4a^2}}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}x=−2ab​±4a2b2−4ac​​=2a−b±b2−4ac​​

This process converts the quadratic into vertex form a(x+m)2+na(x+m)^2+na(x+m)2+n, where:

• m=b2am=\frac{b}{2a}m=2ab​
• n=c−b24an=c-\frac{b^2}{4a}n=c−4ab2​

Example: Solve x2−4x−3=0x^2-4x-3=0x2−4x−3=0 by completing the square:

• Move constant:

x2−4x=3x^2-4x=3x2−4x=3

• Half of −4-4−4 is −2-2−2, square it: 444
• Add 4 to both sides:

x2−4x+4=3+4x^2-4x+4=3+4x2−4x+4=3+4

• Left side is (x−2)2(x-2)^2(x−2)2, right side is 7:

(x−2)2=7(x-2)^2=7(x−2)2=7

• Take square root:

x−2=±7x-2=\pm \sqrt{7}x−2=±7​

• Solve for xxx:

x=2±7x=2\pm \sqrt{7}x=2±7​

This gives the two solutions x=2+7x=2+\sqrt{7}x=2+7​ or x=2−7x=2-\sqrt{7}x=2−7​

. This method is useful for solving quadratic equations and rewriting them in vertex form to analyze their graphs.