Solving quadratic equations by completing the square

When a quadratic equation cannot be factorized, we can use the method of completing the square to solve the equation.

(a + b)^2 = a^2 + 2ab + b^2

(a - b)^2 = a^2 - 2ab + b^2

4-step approach:
1. isolate X's on one side of the equation
2. factor out the leading coefficient of

X^2

3. "completing the square"
• X-side: inside the bracket, add (half of the coefficient of

X)^2

• Y-side: add [ leading coefficient

\cdot

(half of the coefficient of

X)^2

]
4. clean up
• X-side: convert to perfect-square form
• Y-side: clean up the algebra

1.
Solving a quadratic equation with TWO REAL SOLUTIONS
Solve by completing the square: $x^2+10x+6=0$

2.
Solving a quadratic equation with ONE (REPEATED) REAL SOLUTION
Solve by completing the square: $9x^2+25=30x$

3.
Solving a quadratic equation with TWO COMPLEX SOLUTIONS
Solve by completing the square: $-3x^2-24x=49$