Last updated on July 7th, 2025
To understand and solve quadratic equations, we can use the method of completing the square. This method is used to convert a quadratic expression into vertex form. For instance, this technique transforms a quadratic expression of the form ax2 + bx + c into the vertex form, which is a(x - h)2 + k, where (h, k) represents the vertex of the parabola. Hence, the left-hand side becomes a perfect square trinomial, helping in rewriting it in vertex form. In this article, we will explore completing the square in detail.
In algebra, completing the square is a technique used to rewrite a quadratic expression in a form that is a perfect square. For example, a quadratic equation like:
ax2 + bx + c = 0
It can be rewritten (by completing the square) in the form:
a (x + p)2 + q = 0
Where p and q are numbers we calculate during the process.
We can use the process of completing the square to find the roots or zeros of a quadratic equation or quadratic polynomial, and to factorize the equation. When the expression is not possible to factorize, this technique plays an important role.
For instance, x2 + 2x + 3 cannot be factorized using real numbers, because there are no two numbers that add to 2 and multiply to 3. In such cases, we use completing the square to rewrite it in a new form:
a (x + m)2 + n
This form helps us express the quadratic as a perfect square plus a constant.
Here, by rewriting the expression as (x + m), we complete the square.
We have to follow certain steps for the method of completing the square.
Step 1: Express the quadratic equation as x2 + bx + c.
Here, 1 must be the coefficient of x2. If it is not 1, the number will be a common factor and placed outside.
Step 2: Find the half of the coefficient of x.
Step 3: Find the square of the number (half of the coefficient).
Step 4: Add and subtract the square within the expression to maintain equality.
Step 5: To complete the square, factorize the polynomial and use the algebraic identity.
x2 + 2xy + y2 = (x + y)2
Or
x2 - 2xy + y2 = (x - y)2
Now, let us take an example for better understanding.
Complete the square for -4x2 - 8x - 12.
Step 1: First, we need to find if the coefficient is 1. Here, the coefficient of x2 is not 1, so the number (-4) is placed outside as a common factor.
-4x2 - 8x - 12 = -4 (x2 + 2x + 3)
However, the coefficient of x2 is now 1.
Step 2: Next, we need to find half of the coefficient of x.
Coefficient of x = 2
Half of 2 = 1
Step 3: Find the square of 1.
12 = 1
Step 4: Add and subtract the square to the x2 term.
-4 (x2 + 2x + 3) = -4 (x2 + 2x + 1 - 1 + 3)
Step 5: Factorize the polynomials and apply the algebraic identity.
x2 + 2xy + y2 = (x + y)2
Here, the first three terms: x2 + 2x + 1
Last two terms: -1 + 3
x2 + 2x + 1 = (x + 1)2
-4 (x2 + 2x + 1 - 1 + 3) = -4 ( (x + 1)2 - 1 + 3
Next, simplify the last two terms: -1 + 3
-1 + 3 = 2
Now, the expression becomes:
-4 ( ( x + 1)2 + 2
Next, we can distribute the -4:
-4 (x + 1)2 - 8
Hence, the final result is:
-4x2 - 8x - 12 = -4 (x + 1)2 - 8
Now, this is the completed square form:
a (x + m)2 + n
Converting a quadratic equation or polynomial into a perfect square with an extra constant is known as the process of completing the square.
For a quadratic expression: ax2 + bx + c = 0
Here, a, b, and c are real numbers, but a is not equal to 0.
The formula for completing the square is:
ax2 + bx + c a (x + m)2 + n
Here,
m = b / 2a
n = c - (b2 / 4a)
When completing the square of a given expression, we first need to find the values of m and n, then substitute these values into the formula.
Now, let us understand the application of the formula for completing the square.
For example, x2 - 4x - 8 = 0
Now, we can apply the formula for completing the square:
ax2 + bx + c a (x + m)2 + n
Here, a = 1
b = -4
c = -8
Now, we can find the value of m and n.
m = b / 2a = -4 / 2 (1) = -2
n = c - b2 / 4a = -8 - (-4)2 / 4 (1)
n = -8 - 16 / 4 = -8 - 4
n = -12
So, the expression becomes:
x2 - 4x - 8 = (x - 2)2 - 12
Now, we can solve the expression.
(x - 2)2 - 12 = 0
(x - 2)2 = 12
x - 2 = √12 = 2√3
x = 2 ∓ 2√3
Therefore, x = 2 + 2√3 or x = 2 - 2√3
Next, divide the rectangle into two equal parts. So, b / 2a will be the length of each rectangle.
Fix one rectangle to the right side of the square and the next one to the bottom of the square.
To complete the geometric square, we need to add a square of area [ (b / 2a)2 ] to x2 + (b / a) x. To retain the value of the expression, we must subtract it. Thus, to complete the square:
x2 + (b / a) x = x2 + (b / a) x + (b / 2a)2 - (b / 2a)2
x2 + (b / a) x = x2 + (b / a) x + (b / 2 a)2 - b2 / 4a2
Multiplying and dividing (b / a) x with 2 gives,
x2 + (2x × b / 2a) + (b / 2a)2 - b2 / 4a2
Now, we can use the identity, x2 + 2xy + y2 = (x + y)2
The equation above can be expressed as,
x2 + (b / a) x = (x + b / 2a)2 - (b2 / 4a2)
Now, we had:
ax2 + bx + c = a (x2 + b / a x) + c
Next, substitute the completed square form:
= a ( (x + b / 2a)2 - b2 / 4a2) + c
Now, distribute the ‘a’:
a (x + b / 2a)2 - a (b2 / 4a2) + c
Then, simplify:
a (b2 / 4a2) = b2 / 4a
Therefore, the expression becomes:
a (x + b / 2a)2 - b2 / 4a + c
Next, arrange the constants:
= a (x + b / 2a)2 - (c - b2 / 4a)
Thus, ax2 + bx + c = a (x + b / 2a)2 - (c - b2 / 4a)
The expression follows the form:
a (x + m)2 + n
Where, m = b / 2a and n = c - b2 / 4a
Sometimes, completing the square seems tricky for students when they solve quadratic equations. Thus, they often make some errors that lead them to incorrect answers. Here are some common mistakes and their helpful solutions to prevent them.
Understanding the significance of the method of completing the square is useful since it can be applied in various situations. Here are some real-world applications of the concept.
Complete the square of: x^2 + 8x
(x + 4)2 - 16
Here, the given expression is x2 + 8x
First, we need to find the half of the coefficient of x:
8 / 2 = 4
Then, square it:
42 = 16
Now, add and subtract 16:
x2 + 8x + 16 - 16
Here, we take the first three terms:
x2 + 8x + 16
We want to express it in the form:
(x + a)2
(x + a)2 = x2 + 2ax + a2
In the given expression, x2 + 8x + 16
The coefficient of x is 8.
That corresponds to 2a.
2a = 8
a = 4
Therefore, a2 = 42 = 16
So, x2 + 8x + 16 = (x + 4)2
Hence, the expression becomes:
(x + 4)2 - 16
Thus, x2 + 8x = (x + 4)2 - 16
Solve the equation, x^2 - 10x
(x - 5)2 - 25
The given expression is x2 - 10x
Half of -10 = -5
Square of -5 (-52) = 25
Next, add and subtract 25:
x2 - 10x + 25 - 25
Rewrite it as:
(x - 5)2 - 25 = (x - 5)2
Hence, the expression becomes:
(x - 5)2 - 25
Thus, x2 - 10x = (x - 5)2 - 25
Solve x^2 + 7x + 5 = 0
(-7 ± √29) / 2
The given expression is:
x2 + 7x + 5
Now, move the constant to the right side.
x2 + 7x = - 5
Take the half of the coefficient of x:
The coefficient of x is 7.
Half of 7 is:
7 / 2
Next, square it:
(7 / 2)2 = 49 / 4
So, we need to add 49 / 4 to both sides of the equation.
x2 + 7x + 49 / 4 = -5 + 49 / 4
Next, convert -5 to a fraction with denominator 4:
-5 = -20 / 4
Now, add:
-20 / 4 + 49 / 4 = 29 / 4
Hence, the expression is:
x2 + 7x + 49 / 4 = 29 / 4
Here, the left-hand side is a perfect square trinomial:
x2 + 7x + 49 / 4 = (x + 7 / 2)2
Rewrite the equation:
(x + 7 / 2)2 = 29 / 4
Now, take the square root of both sides:
x + 7 / 2 = √29 / 4
x + 7 / 2 = √29 / 2
Next, subtract 7 / 2 from both sides:
x = - 7 / 2 ± √29 / 2
Since the denominators are the same, we can write it as:
x = -7 ± √29 / 2
Complete the square of the expression: 2x^2 + 12 x
2 (x + 3)2 - 18
First, we need to factor out 2.
2 (x2 + 6x)
Half of 6 = 3
Square it:
32 = 9
Add and subtract inside:
2 (x2 + 6x + 9 - 9)
Rewrite it as:
2 ( (x + 3)2 - 9) = 2 (x + 3)2 - 18
Thus, 2x2 + 12x = 2 (x + 3)2 - 18
Complete the square of the expression: x^2 - 5x
(x - 5 / 2)2 - 25 / 4
The given expression is:
x2 - 5x
Half of -5 = -5 / 2
Square it:
(-5 / 2)2 = (25 / 4)
We add and subtract the square inside the expression so the value does not change:
x2 - 5x = x2 - 5x + 25 / 4 - 25 / 4
Now, we apply the identity:
x2 - 2xy + y2 = (x - y)2
Here, x2 - 5x + 25 / 4 is a perfect square trinomial.
So, it can be written as:
(x - 5 / 2)2
Hence the expression becomes:
x2 - 5x = (x - 5 / 2)2 - 25 / 4
Therefore, the answer is:
x2 - 5x = (x - 5 / 2)2 - 25 / 4
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
: He loves to play the quiz with kids through algebra to make kids love it.