Kid-friendly explanation
Completing the square is like turning a messy shape into a perfect square so it is easier to work with. In algebra, you change an equation like a puzzle so one side becomes something squared, such as (x + 3)2. Once it looks like that, you can take square roots and solve. It is one more tool to help you solve quadratic equations.
Worked examples
Example 1 (Algebra 1 level): Solve a quadratic by completing the square
Solve the equation x2 + 6x + 5 = 0 by completing the square.
Step 1: Move the constant term (the number without x) to the other side.
x2 + 6x + 5 = 0 becomes x2 + 6x = -5.
Step 2: Take half of the coefficient of x, then square it. The coefficient of x is 6. Half of 6 is 3, and 32 = 9.
Step 3: Add this square to both sides to keep the equation balanced.
x2 + 6x + 9 = -5 + 9, so x2 + 6x + 9 = 4.
Step 4: Rewrite the left side as a perfect square. x2 + 6x + 9 = (x + 3)2.
Now the equation is (x + 3)2 = 4.
Step 5: Take the square root of both sides.
x + 3 = 2 or x + 3 = -2.
Step 6: Solve each simple equation.
If x + 3 = 2, then x = -1. If x + 3 = -2, then x = -5.
So the solutions are x = -1 and x = -5.
Example 2 (More advanced: leading coefficient not 1 and vertex form)
Rewrite the quadratic function f(x) = 2x2 - 8x + 3 in vertex form by completing the square, then identify the vertex.
Step 1: Factor out the coefficient of x2 from the x terms.
f(x) = 2x2 - 8x + 3 becomes f(x) = 2(x2 - 4x) + 3.
Step 2: Complete the square inside the parentheses. Take half of -4, which is -2, and square it: (-2)2 = 4.
Add and subtract 4 inside the parentheses so the overall value does not change:
f(x) = 2(x2 - 4x + 4 - 4) + 3.
Step 3: Group the perfect square trinomial and the leftover constant inside the parentheses.
f(x) = 2[(x2 - 4x + 4) - 4] + 3.
Rewrite x2 - 4x + 4 as (x - 2)2:
f(x) = 2[(x - 2)2 - 4] + 3.
Step 4: Distribute the 2 and simplify the constants.
f(x) = 2(x - 2)2 - 8 + 3 = 2(x - 2)2 - 5.
This is vertex form, f(x) = a(x - h)2 + k, where a = 2, h = 2, and k = -5.
The vertex of the parabola is (2, -5).
Going deeper with completing the square
Completing the square is one of the main methods for solving quadratic equations, along with factoring, taking square roots directly, and using the quadratic formula. It is especially helpful when a quadratic does not factor nicely with whole numbers.
This method also connects algebra to graphing. When a quadratic function is written in vertex form using completing the square, the vertex and direction of opening of the parabola are easy to see, which helps interpret real-world situations modeled by quadratics.
In more advanced work, completing the square is used to rewrite equations like ax2 + by2 + cx + dy + e = 0 into standard forms. This helps classify graphs such as circles and ellipses and understand their key features.
Common mistakes with completing the square
- Mistake: Adding the square number to only one side of the equation, for example turning x2 + 4x = 5 into x2 + 4x + 4 = 5 instead of x2 + 4x + 4 = 9.
Correction: Any number added to one side of an equation must also be added to the other side to keep the equation balanced. - Mistake: Forgetting to take half of the x coefficient before squaring, such as using 62 = 36 instead of (6 ÷ 2)2 = 9 when the coefficient is 6, which leads to a wrong perfect square and wrong solutions.
Correction: Always divide the x coefficient by 2 first, then square that result to find the number to add. - Mistake: When a leading coefficient is factored out, forgetting that the added square inside parentheses is multiplied by that coefficient, for example in 2(x2 - 4x) adding 4 only to the right side instead of 2 × 4 = 8, which gives an incorrect equation.
Correction: If you complete the square inside parentheses with a factor outside, adjust the other side of the equation by the factor times the added inside number.
How completing the square shows up in everyday life
Completing the square itself is an algebra tool, but the problems it helps solve show up in daily life. Quadratic equations can model the height of a ball in the air, the shape of a satellite dish, or the path of water from a fountain. Completing the square helps find maximum or minimum values in these situations, such as the highest point of a jump or the best dimensions for an area.