10.2 Solve Quadratic Equations by Completing the Square

Completing the square is a technique for solving quadratic equations that works every time, even when factoring doesn't. It transforms an expression like $x^2 + bx$ into a perfect square trinomial, which you can then factor neatly and solve with square roots.

This method is worth learning well because it's the foundation for deriving the quadratic formula, and it also helps you find the vertex of a parabola directly from the equation.

Completing the Square

Binomial to Perfect Square Trinomial

The core idea is to take a binomial like $x^2 + bx$ and figure out what constant to add so it becomes a perfect square trinomial you can factor.

Here's the rule: take half the coefficient of $x$, then square it. That gives you the number to add.

• Half the coefficient of $x$ is $\frac{b}{2}$
• Square it to get $\left(\frac{b}{2}\right)^2$
• The result: $x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2$

Example: For $x^2 + 6x$, half of 6 is 3, and $3^2 = 9$. So $x^2 + 6x + 9 = (x + 3)^2$.

Example: For $x^2 - 10x$, half of $-10$ is $-5$, and $(-5)^2 = 25$. So $x^2 - 10x + 25 = (x - 5)^2$.

Notice that the sign inside the parentheses matches the sign of $\frac{b}{2}$.

Completing the Square with Coefficient 1

When the coefficient of $x^2$ is already 1, you can solve $x^2 + bx + c = 0$ with these steps:

1. Move the constant to the other side. Subtract $c$ from both sides to get $x^2 + bx = -c$.
2. Complete the square. Calculate $\left(\frac{b}{2}\right)^2$ and add it to both sides: $x^2 + bx + \left(\frac{b}{2}\right)^2 = -c + \left(\frac{b}{2}\right)^2$
3. Factor the left side into $\left(x + \frac{b}{2}\right)^2$.
4. Take the square root of both sides. Don't forget the $\pm$: $x + \frac{b}{2} = \pm\sqrt{-c + \left(\frac{b}{2}\right)^2}$
5. Solve for $x$ by subtracting $\frac{b}{2}$ from both sides.

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

1. Move the constant: $x^2 + 6x = -5$
2. Half of 6 is 3, and $3^2 = 9$. Add 9 to both sides: $x^2 + 6x + 9 = -5 + 9$
3. Factor: $(x + 3)^2 = 4$
4. Square root: $x + 3 = \pm 2$
5. Solve: $x = -3 + 2 = -1$ or $x = -3 - 2 = -5$

A common mistake is adding $\left(\frac{b}{2}\right)^2$ to the left side but forgetting to add it to the right side too. Whatever you do to one side, you must do to the other.

Completing the Square for Any Coefficient

When the leading coefficient $a$ isn't 1, you need one extra step at the start: divide everything by $a$.

1. Divide every term by $a$ so the equation becomes $x^2 + \frac{b}{a}x + \frac{c}{a} = 0$.
2. Move the constant to the other side: $x^2 + \frac{b}{a}x = -\frac{c}{a}$
3. Complete the square. Half of $\frac{b}{a}$ is $\frac{b}{2a}$. Square it to get $\left(\frac{b}{2a}\right)^2$. Add this to both sides.
4. Factor the left side into $\left(x + \frac{b}{2a}\right)^2$.
5. Take the square root of both sides (with $\pm$).
6. Solve for $x$.

Example: Solve $2x^2 + 12x + 10 = 0$

1. Divide by 2: $x^2 + 6x + 5 = 0$
2. Move the constant: $x^2 + 6x = -5$
3. Half of 6 is 3, and $3^2 = 9$. Add 9 to both sides: $(x + 3)^2 = 4$
4. Square root: $x + 3 = \pm 2$
5. Solve: $x = -1$ or $x = -5$

The key thing to remember: you must make the leading coefficient 1 before you complete the square. If you skip this step, the method won't work.

Quadratic Equation Properties

Completing the square connects directly to the graph of a quadratic equation, which is a parabola.

• The vertex is the highest or lowest point on the parabola. When you write the equation in the form $(x + \frac{b}{2})^2 = k$, the vertex is easy to read off.
• The axis of symmetry is a vertical line that passes through the vertex. The parabola is a mirror image on either side of this line.
• The discriminant $b^2 - 4ac$ tells you how many real solutions the equation has:
  • Positive: two distinct real roots (the parabola crosses the x-axis twice)
  • Zero: one repeated real root (the parabola just touches the x-axis at its vertex)
  • Negative: no real roots (the parabola never crosses the x-axis)