Solving by completing the square

Completing the square involves collecting the $x^2$ and $x$ terms together in a bracket which can be quicker than using the quadratic formula to solve a quadratic equation.

The method of completing the square can be streamlined by using the formula:

$$x^2+bx={(x-\frac{b}{2})}^2-{\left(\frac{b}{2}\right)}^2$$

This formula allows you to rewrite the quadratic equation into a form that is easier to solve while keeping the equation balanced.

1. Steps to Solve by Completing the Square:

To solve a quadratic equation using this formula, follow these steps:

1. Ensure the coefficient of $x^2$ is 1. If it is not, divide the entire equation by $a$.
2. Apply the formula: $$x^2+bx={(x-\frac{b}{2})}^2-{\left(\frac{b}{2}\right)}^2$$
3. Substitute this into the original equation and simplify.
4. Solve for $x$ by isolating it using square roots and simplifying.

2. Example: Solve $x^2+6x+5=0$:

Step 1: Ensure the coefficient of $x^2$ is 1:

The coefficient of $x^2$ is already 1, so no adjustment is needed.

Step 2: Apply the formula:

For $x^2+6x$, the formula gives: $$x^2+6x={(x-\frac{6}{2})}^2-{\left(\frac{6}{2}\right)}^2$$ $$x^2+6x=(x-3{)}^2-9$$

Step 3: Substitute into the equation:

Replace $x^2+6x$ in the original equation $x^2+6x+5=0$: $$(x-3{)}^2-9+5=0$$

Simplify: $$(x-3{)}^2-4=0$$

Step 4: Solve for $x$:

Isolate the perfect square: $$(x-3{)}^2=4$$

Take the square root of both sides: $$x-3=\pm \sqrt{4}$$ $$x-3=\pm 2$$

Isolate $x$: $$x=3+2\text{or}x=3-2$$ $$x=5\text{or}x=1$$

3. Special Cases and Notes:

• If the coefficient of $x^2$ is not 1, divide the entire equation by $a$ before applying the formula.
• The method is particularly useful because it avoids the need to manually calculate ${\left(\frac{b}{2}\right)}^2$ separately.
• If the square root results in a negative number, the roots are complex.