The technique of completing the square is a factoring technique that allows us to convert a given quadratic expression or equation in the form ax2+bx+c to the form a(x-h)2+k. We can use this technique to simplify the process of solving equations when we have complex quadratic equations.
In this article, we will look at a summary of the technique of completing the square. Then, we will use this technique to solve some practice problems.
Formula to complete the square
The process of completing the square is used to express a quadratic expression given as $latex ax^2+bx+c$ in the following form:
$latex a(x+p)^2+q$
where p and q are constants.
The simplest case of completing the square happens when we have a=1, that is, the quadratic term has a coefficient equal to 1. In these cases, we have:
$$x^2+bx+c=\left(x+\frac{b}{2}\right)^2-\left(\frac{b}{2}\right)^2+c$$
$$=\left(x+\frac{b}{2}\right)^2-\left(\frac{b^2}{4}\right)+c$$
Completing the square - Step by step method
We can follow the steps below to complete the square of a quadratic expression. This method applies even when the coefficient a is different from 1.
Step 1: If the coefficient a is different from 1, we divide the entire quadratic expression by a to obtain an expression where the quadratic term has a coefficient equal to 1:
$latex x^2+bx+c$
Step 2: We divide the coefficient of x (the coefficient b) by 2:
$$\left(\frac{b}{2}\right)$$
Step 3: We square the expression obtained in step 2:
$$\left(\frac{b}{2}\right)^2$$
Step 4: We add and subtract the expression obtained in step 3 to the expression obtained in step 1:
$$x^2+bx+\left(\frac{b}{2}\right)^2-\left(\frac{b}{2}\right)^2+c$$
Step 5: We factor the quadratic expression by applying the algebraic identity $latex x^2+2xy+y^2=(x+y)^2$:
$$\left(x+\frac{b}{2}\right)^2-\left(\frac{b}{2}\right)^2+c$$
Step 6: We multiply the expression resulting from step 5 by the number by which we divided in step 1.
Solve quadratic equations by completing the square
The method of completing the square allows us to solve quadratic equations easily. When we have a quadratic expression in the form $latex (x-h)^2+k$, we can write it as follows:
$latex (x-h)^2=-k$
Here, we can take the square root of both sides and easily solve for x.
Completing the square - Examples with answers
EXAMPLE 1
Complete the square of the expression $latex x^2+2x-5$.
Solution
Since the coefficient of the quadratic term is equal to 1, we don't have to divide the expression by any numbers initially.
We see that the coefficient b is equal to 2. Therefore, we have:
$$\left(\frac{b}{2}\right)^2=\left(\frac{2}{2}\right)^2=1$$
Adding and subtracting this value, we have:
$$x^2+2x-5=x^2+2x+1-1-5$$
Completing the square and simplifying, we have:
$latex = (x+1)^2-1-5$
$latex = (x+1)^2-6$
EXAMPLE 2
Complete the square of the expression $latex x^2+4x+10$.
Solution
We don't have to apply the first step, since the coefficient of the quadratic term is equal to 1.
Now, we can see that the coefficient b is equal to 4. Therefore, we have:
$$\left(\frac{b}{2}\right)^2=\left(\frac{4}{2}\right)^2$$
$$=2^2$$
When we add and subtract this expression, we have:
$$x^2+4x+10=x^2+4x+2^2-2^2+10$$
Completing the square and simplifying, we have:
$latex = (x+2)^2-4+10$
$latex = (x+2)^2+6$
EXAMPLE 3
Complete the square of the expression $latex 2x^2+6x+6$.
Solution
Here, the expression has a quadratic term with a coefficient other than 1. Therefore, we can divide the entire expression by 2 to get the following
⇒ $latex x^2+3x+3$.
Given that the coefficient b equals 3, we have:
$$\left(\frac{b}{2}\right)^2=\left(\frac{3}{2}\right)^2$$
Adding and subtracting this value, we have:
$$x^2+3x+3=x^2+3x+\left(\frac{3}{2}\right)^2-\left(\frac{3}{2}\right)^2+3$$
Completing the square and simplifying, we have:
$latex = (x+\frac{3}{2})^2-\frac{9}{4}+3$
$latex = (x+\frac{3}{2})^2+\frac{3}{4}$
Since we divided the expression by 2 initially, we multiply the result by 2:
⇒ $latex 2(x+\frac{3}{2})^2+\frac{3}{2}$
EXAMPLE 4
Solve the equation $latex x^2+4x-5=0$ using the method of completing the square.
Solution
In this equation, b is equal to 4. Therefore, we have:
$$\left(\frac{b}{2}\right)^2=\left(\frac{4}{2}\right)^2$$
$$=2^2$$
Adding and subtracting this value to the quadratic equation, we have:
$$x^2+4x-5=x^2+4x+2^2-2^2-5$$
Completing the square and simplifying, we have:
$latex = (x+2)^2-4-5$
$latex = (x+2)^2-9$
Now, we can write the equation as follows:
⇒ $latex (x+2)^2=9$
Taking the square root of both sides, we have:
⇒ $latex x+2=\sqrt{9}$
⇒ $latex x+2=3$
⇒ $latex x=1$
EXAMPLE 5
Solve the equation $latex 2x^2-8x-8=0$ using the method of completing the square.
Solution
We divide the equation by 2 to obtain an equation where the coefficient of the quadratic term is equal to 1:
$latex x^2-4x-4=0$
Now, we see that the coefficient b is equal to -4. Therefore, we have:
$$\left(\frac{b}{2}\right)^2=\left(\frac{-4}{2}\right)^2$$
$$=(-2)^2$$
Adding and subtracting that value to the equation, we have:
$$x^2-4x-4=x^2-4x+(-2)^2-(-2)^2-4$$
Completing the square and simplifying, we have:
$latex = (x-2)^2-4-4$
$latex = (x-2)^2-8$
Now, we write the equation as follows:
⇒ $latex (x-2)^2=8$
Taking the square root of both sides, we have:
⇒ $latex x-2=\sqrt{8}$
⇒ $latex x=2\pm \sqrt{8}$
EXAMPLE 6
Find the solutions of the equation $latex 2x^2+12x-14=0$ using the method of completing the square.
Solution
Dividing the equation by 2, we can make the coefficient of the quadratic term equal to 1:
⇒ $latex x^2+6x-7=0$
Now, we have that the coefficient b is equal to 6. Therefore, we have:
$$\left(\frac{b}{2}\right)^2=\left(\frac{6}{2}\right)^2$$
$$=3^2$$
If we add and subtract this value to the equation, we have:
$$x^2+6x-7=x^2+6x+3^2-3^2-7$$
Completing the square and simplifying, we have:
$latex = (x+3)^2-9-7$
$latex = (x+3)^2-16$
We can write the equation as follows:
$latex (x+3)^2=16$
Taking the square root of both sides, we have:
⇒ $latex x+3=4$
⇒ $latex x=1$
EXAMPLE 7
Solve the quadratic equation $latex 3x^2-12x-3=0$ using the method of completing the square.
Solution
We start by dividing the equation by 3 to make the coefficient of the quadratic term equal to 1:
⇒ $latex x^2-4x-1=0$
We see that the coefficient b is equal to -4. Therefore, we have:
$$\left(\frac{b}{2}\right)^2=\left(\frac{-4}{2}\right)^2$$
$$=(-2)^2$$
Adding and subtracting this value to the equation, we have:
$$x^2-4x-1=x^2-4x+(-2)^2-(-2)^2-1$$
Completing the square and simplifying, we have:
$latex = (x-2)^2-4-1$
$latex = (x-2)^2-5$
We can write the equation as follows:
$latex (x-2)^2=5$
We can solve the equation by taking the square root of both sides:
⇒ $latex (x-2)=\sqrt{5}$
⇒ $latex x=2\pm \sqrt{5}$
Completing the square - Practice problems
See also
Interested in learning more about completing the square? Take a look at these pages:
