Lesson 4: Completing the Square

New Concept

"Completing the square" is a powerful method for solving any quadratic equation. It transforms one side of the equation into a perfect square trinomial, $(x+p)^2$, allowing you to solve it easily by taking the square root of both sides.

What’s next

You've got the foundational concept. Next, you'll master this skill through interactive examples and practice cards that break down the step-by-step solving process.

Property

The square of a binomial is a quadratic trinomial.

The coefficient of the linear term, $2p$, is twice the constant in the binomial, and the constant term of the trinomial, $p^2$, is its square.

Examples

• The square of $(x+5)$ is $x^2 + 2(5)x + 5^2$, which simplifies to $x^2 + 10x + 25$.
• The square of $(x-3)$ is $x^2 + 2(-3)x + (-3)^2$, which simplifies to $x^2 - 6x + 9$.
• The square of $(x-12)$ is $x^2 + 2(-12)x + (-12)^2$, which simplifies to $x^2 - 24x + 144$.

Explanation

Expanding a binomial squared always creates a predictable three-term pattern. Recognizing this pattern is the first step to reversing the process, which allows us to factor complex expressions into a simple, squared form.

Property

To complete the square for the expression $x^2 + bx$, we add a constant term. We can find this constant term by taking one-half the coefficient of $x$ and then squaring the result.

1. Identify the coefficient of the linear term, $b$.
2. Calculate $p = \frac{1}{2}b$.
3. Add $p^2$ to the expression.

Examples

• For $x^2 - 12x$, half of $-12$ is $-6$, and $(-6)^2 = 36$. Adding 36 gives $x^2 - 12x + 36 = (x-6)^2$.
• For $x^2 + 5x$, half of $5$ is $\frac{5}{2}$, and $(\frac{5}{2})^2 = \frac{25}{4}$. Adding $\frac{25}{4}$ gives $x^2 + 5x + \frac{25}{4} = (x + \frac{5}{2})^2$.
• For $x^2 - 18x$, half of $-18$ is $-9$, and $(-9)^2 = 81$. Adding 81 gives $x^2 - 18x + 81 = (x-9)^2$.

Explanation

Completing the square is a technique to turn an expression like $x^2+bx$ into a perfect square trinomial. By adding the 'magic number' $(\frac{b}{2})^2$, you create a new expression that can be easily factored into a binomial squared.

Property

To solve a quadratic equation of the form $x^2+bx+c=0$:

1. Move the constant term to the other side: $x^2+bx = -c$.
2. Complete the square on the left. Add $p^2 = (\frac{b}{2})^2$ to both sides of the equation: $x^2+bx+p^2 = -c+p^2$.
3. Write the left side as a binomial squared: $(x+p)^2 = -c+p^2$.
4. Use extraction of roots to find the solutions.

Examples

• To solve $x^2 - 6x - 7 = 0$, first write $x^2-6x=7$. Add $(\frac{-6}{2})^2=9$ to both sides: $x^2-6x+9=7+9$, so $(x-3)^2=16$. The solutions are $x=7$ and $x=-1$.
• To solve $x^2 - 4x - 3 = 0$, first write $x^2-4x=3$. Add $(\frac{-4}{2})^2=4$ to both sides: $x^2-4x+4=3+4$, so $(x-2)^2=7$. The solutions are $x = 2 \pm \sqrt{7}$.
• To solve $x^2+9x+20=0$, first write $x^2+9x=-20$. Add $(\frac{9}{2})^2=\frac{81}{4}$ to both sides: $x^2+9x+\frac{81}{4}=-20+\frac{81}{4}$, so $(x+\frac{9}{2})^2=\frac{1}{4}$. The solutions are $x=-4$ and $x=-5$.

Explanation

This method transforms any quadratic equation into the form $(x+p)^2=q$. By forcing one side to be a perfect square, we can easily solve for $x$ by taking the square root of both sides, which is often simpler than factoring.

Property

To Solve a Quadratic Equation by Completing the Square:

1. Write the equation in standard form. Divide both sides of the equation by the coefficient of the quadratic term, and subtract the constant term from both sides.
2. Complete the square on the left side: Multiply the coefficient of the first-degree term by one-half, then square the result. Add this value to both sides of the equation.
3. Write the left side of the equation as the square of a binomial. Simplify the right side.
4. Use extraction of roots to finish the solution.

Examples

• To solve $2x^2 - 6x - 5 = 0$, first divide by 2 to get $x^2 - 3x - \frac{5}{2} = 0$. This becomes $x^2-3x = \frac{5}{2}$. Completing the square gives $(x - \frac{3}{2})^2 = \frac{19}{4}$, so $x = \frac{3}{2} \pm \sqrt{\frac{19}{4}}$.
• To solve $3x^2+12x+2=0$, first divide by 3 to get $x^2+4x+\frac{2}{3}=0$. This becomes $x^2+4x = -\frac{2}{3}$. Completing the square gives $(x+2)^2 = \frac{10}{3}$, so $x = -2 \pm \sqrt{\frac{10}{3}}$.
• To solve $-4x^2 - 36x - 65 = 0$, first divide by $-4$ to get $x^2 + 9x + \frac{65}{4} = 0$. This becomes $x^2+9x = -\frac{65}{4}$. Completing the square gives $(x+\frac{9}{2})^2=4$, so $x = -\frac{5}{2}$ and $x = -\frac{13}{2}$.

Explanation

If the $x^2$ term has a coefficient other than 1 (like $ax^2$), you cannot complete the square directly. The essential first step is to divide the entire equation by 'a', then you can proceed with the standard steps.