Completing the Square

What is Completing the Square?

Completing the square is a method to solve quadratic equations by rewriting them into a form that makes solving easier. It’s particularly useful for finding the vertex of a parabola or solving quadratics that don’t factorise neatly.

The standard quadratic equation is: $$ax^2 + bx + c = 0$$

When completing the square, we rewrite it as: $$a(x + h)^2 + k = 0$$

where $h$ and $k$ are numbers you calculate

Steps to Complete the Square

1. Make the coefficient of $x^2$ equal to 1:

   • If $a \not = 1$, divide the entire equation by $a$

2. Rewrite the quadratic:

   • Focus on the $x^2$ and $x$ terms, leaving the constant aside for now
3. Add and subtract a perfect square:
   • Take half of the coefficient of $x$, square it, and add and subtract it in the equation
4. Factorise the perfect square:
   • The $x^2$ and $x$ terms should now form a perfect square trinomial, which you can write as $(x + p)^2$
5. Simplify the equation:
   • Move constants around to solve for $x$

 

Examples

Example 1

Solve $x^2 + 6x + 5 = 0$ by completing the square

1. Rewrite the equation: $$x^2 + 6x + 5 = 0$$
2. Focus on the $x^2 + 6x$ part: Take half of the coefficient of $x$, square it: $$\text{Half of} 6 \text{ is } 3, \text{ and } 3^2 = 9$$
3. Add and subtract 9: $$x^2 + 6x + 9 - 9 + 5 = 0$$
4. Factorise the perfect square: $$(x + 3)^2 - 9 + 5 = 0$$
5. Simplify: $$(x + 3)^2 - 4 = 0$$
6. Solve for $x$: $$(x + 3)^2 = 4 \\ \text{Take the square root of both sides:}  \\ x + 3 = \pm 2  \\  \text{Solve for } x \\  x = -3 + 2 = -1  \text{ or }  x = -3 -2 = -5$$

Solution: $$x = -1 \quad \text{or} \quad x = -5$$

 

 

 

Example 2

Solve $2x^2 + 8x -3 =0$ by completing the square

1. Make the coefficient of $x^2$ equal to 1: Divide the whole equation by 2: $$x^2 + 4x - \frac{3}{2} = 0$$
2. Focus on $x^2 + 4x$: Take half of the coefficient of $x$, square it: $$\text{Half of } 4 \text{is } 2, \text{and } 2^2 = 4$$
3. Add and subtract 4: $$x^2 + 4x + 4 -4 - \frac{3}{2} = 0$$
4. Factorise the perfect square : $$(x + 2)^2 - 4 - \frac{3}{2} = 0$$
5. Simplify constants: Combine $-4 - \frac{3}{2} = 0$: $$(x +2)^2 - \frac{11}{2} = 0$$
6. Solve for $x$: $$(x + 2)^2 = \frac{11}{2}  \\  \text{Take the square root: } \\ x + 2 = \pm \sqrt{\frac{11}{2}} \\ \text{Solve for } x \\ x = -2 \pm \sqrt{\frac{11}{2}}$$

Solution: $$x = -2 + \sqrt{\frac{11}{2}} \quad \text{or} \quad x = -2 - \sqrt{\frac{11}{2}}$$