Completing Squares

In mathematics, we generally talk about completing the square when a quadratic equation has coefficients that are not perfect squares. For example, the quadratic equation

x² + 5x + 6 = 0

square can be completed to  

(a+½)(a+1)=0

which can be written as

ax²+2ax+2=0

This equation has a solution of x=2.

To break it down more, completing the square means first transforming an algebraic expression so that both terms are perfect squares (squares of numbers with no factors). This is done by multiplying each term by the negative of the other and adding them.

What does Completing Square mean?

A completing square is a math problem that involves adding the same number to both sides of an equation so that the calculations are balanced.

For example:

(+1)2 = 2 + 1 + 2 = 5

(+3)2 = 6 + 3 + 9 = 18

You could say (1)2= 1 or (5)2 would equal 25.

All quadratic equations can be solved by completing squares, and what I mean by “solved” is that you can turn the given quadratic equation into an x and y number.

The Square Method:

To complete a quadratic equation using the square method, you will first want to get a pair of binomials with coefficients that aren’t perfect squares.

Then you want to add the same number (not necessarily real) to both the numerator and denominator of the numerical coefficients.

This will create a quadratic equation with perfect squares in it.

If you look at the equation, it will have perfect squares on both sides of the equation.

You can now complete the square by taking half of the coefficient next to an “x” and adding it to both sides of the equation.

Then take half of the coefficient next to a “y” and subtract it from both sides.

If you did this right, you should now have a zero in your quadratic equation.

• Rewrite the equation in the form a² + b² = c²
• Divide each side of the equation by a
• Find pairs of numbers (x,y) that make each half of the equation equal to c.
• Put these points into the formula to find x and y

Steps of Completing the Square Method 

Completing the square is the method to find the solutions to quadratic equations. It is widely used to find the zeros of any quadratic equation. The solution of any quadratic equation is also called its zeros. The zeros of any equation are those values that provide the zero after putting in the place of variables of the equation. If there is any constant value written in equals to the equation, then the same constant will come as a result after putting the zeros of the quadratic equation. Let’s learn about the steps of solving the quadratic equation:

Suppose the quadratic equation be: 

ax2 + bx + c = 0, 

Here, a is not equal to 1.

While using the completing the square method, we keep the equation in the written below form:

(x ± h)2 = d 

In the above equation, d and h are the constants. 

Step 1: In this step, divide all the terms of the quadratic equation with the coefficient value of x2

Step 2: Compute b/2.

Step 3: Compute the square of b/2, that is (b/2)2. 

Step 4: In this step, add the resulting value of (b/2)2 on both sides of the given quadratic equation. 

Step 5:  In this step, complete the square of the left-hand side. The left-hand side terms get factored using the identities of, squaring the terms with two variables. The example of such identity is, (x + y)2 = x2 + 2xy + y2. 

Step 6: After acquiring the square term from the 5th step, isolate this term from the whole equation by changing the sides of the other remaining terms. 

Step 7: In the last step, take the square root of both sides and then solve it. After solving the square root, the result will be the “solution or zero” of the quadratic equation.

Example of completing the square: 3x² – 15x + 4 = 0 

• Rewrite the equation in the form a² + b² = c²  3x² – 15x + 4 = 0
• Divide each side of the equation by a 
• Find pairs of numbers (x,y) that make each half of the equation equal to c. 
• Put these points into the formula to find x and y: (1)(3)(0)(4) = 0 

There are two answers because we have 0 as our base number. So we can just put these numbers in our formula and solve them.

Conclusion

Completing the square is a useful method for completing a quadratic equation. If you can use a complete square, you will be able to write a quadratic equation in the form ax² + bx + c = 0, which is the form that it is typically required to have. By completing the square, you are essentially adding a number to both sides of the equation to equal balance it. The notation “completing square” can also refer to generating a Pythagorean triple using this method.

Completing squares can be used to find the solution of quadratics and other types of equations that have three terms.