Lesson 104: Solving Quadratic Equations by Completing the Square

New Concept

Complete the square of $x^2 + bx$ by adding $(\frac{b}{2})^2$ to the expression.

What’s next

Next, you will apply this method to solve quadratic equations by creating a perfect square and then taking the square root of both sides.

Property

To complete the square for an expression like $x^2 + bx$, you add a special value: $(\frac{b}{2})^2$. This action transforms the expression into a perfect-square trinomial, which factors neatly into $(x + \frac{b}{2})^2$ and makes it easier to handle.

Explanation

This method is like a puzzle! You are finding the one missing piece that turns your expression into a perfect, balanced square. This makes it way easier to work with when you are solving the full quadratic equation later on. It’s a foundational step for solving quadratics!

Examples

• $x^2 + 12x \rightarrow x^2 + 12x + (\frac{12}{2})^2 = x^2 + 12x + 36 = (x+6)^2$
• $y^2 - 10y \rightarrow y^2 - 10y + (\frac{-10}{2})^2 = y^2 - 10y + 25 = (y-5)^2$

Property

For an equation in the form $x^2+bx=c$, first complete the square by adding $(\frac{b}{2})^2$ to both sides. Then, factor the left side into a binomial square and solve by taking the square root of both sides.

Explanation

After you have completed the square, you get a simple $(x+k)^2=d$ format. Now you can free 'x' from its parentheses prison by taking the square root! Just do not forget the $\pm$ symbol, which gives you two possible answers.

Examples

• $x^2 + 6x = 16 \rightarrow (x+3)^2 = 16+9 \rightarrow (x+3)^2 = 25 \rightarrow x+3=\pm5$, so $x=2$ or $x=-8$.
• $x^2 - 2x = 8 \rightarrow (x-1)^2 = 8+1 \rightarrow (x-1)^2 = 9 \rightarrow x-1=\pm3$, so $x=4$ or $x=-2$.

Let's transform this equation into a perfect square to make solving it a breeze. This first example shows the core technique of completing the square when the quadratic term is already simplified.

Solve the equation by completing the square: $x^2 + 12x = 13$.

1. First, we need to complete the square on the left side. We add the square of half the coefficient of the $x$-term to both sides.

2. Simplify the term in the parentheses.

3. Simplify the expression.

4. Factor the perfect-square trinomial on the left and simplify the right side.

5. Now, take the square root of both sides to solve for $x$.

6. Simplify the equation.

7. Write the two possible equations and solve each one.

Property

When solving $ax^2 + bx = c$, you must first divide every term in the equation by the coefficient 'a'. This crucial first step ensures the $x^2$ coefficient is equal to 1, allowing you to proceed with completing the square as you normally would.

Explanation

Do not let a number in front of $x^2$ scare you. Your first mission is to eliminate it by dividing the whole equation by that number. Once the $x^2$ term is standing alone, you are back on familiar ground and can solve the problem.

Examples

• $2x^2 + 8x = 10 \rightarrow x^2 + 4x = 5 \rightarrow (x+2)^2 = 5+4 \rightarrow (x+2)^2 = 9 \rightarrow x=1$ or $x=-5$.
• $3x^2 - 12x = 15 \rightarrow x^2 - 4x = 5 \rightarrow (x-2)^2 = 5+4 \rightarrow (x-2)^2 = 9 \rightarrow x=5$ or $x=-1$.