Completing the Square when a = 1

A quadratic equation is an equation that contains a squared variable as its highest power on any variable. The general form of a quadratic equation is:

ax2 + bx + c = 0


Where a, b, and c are constants and a ≠ 0. In other words there must be a x2 term.

Some examples are:

x2 + 3x - 3 = 0
4x2 + 9 = 0 (Where b = 0)
x2 + 5x = 0 (where c = 0)

One way to solve a quadratic equation is by completing the square.

ax2 + bx + c = 0 → (x- r)2 = S


Where r and s are constants.

This discussion will focus on completing the square when a, the x2 -coefficient, is 1.

Let's solve the following equation by completing the square: x2 = 6x - 7

Step 1: Write the equation in the general form

ax2 + bx + c = 0.

x2 - 6x + 5 = 0

Step 2: Move c, the constant term, to the right-hand side of the equation.

c = 5


x2 - 6x = -5

Step 3: Divide b, the x-coefficient, by two and square the result.

b = -6


6 2 =3r


(-3)2 = 9

Step 4: Add the result from Step 3 to both sides of the equation.

x2 - 6x + 9 = -5 + 9

Step 5: Rewrite the left-hand side as a perfect square and simplify the right-hand side.


When rewriting in perfect square format the value in the parentheses is the b, x-coefficient, divided by 2 as found in Step 3.

(x - 3)2 = -5 + 9


(x - 3)2 = 4

Now that the square has been completed, solve for x.

Step 6: Take the square root of both sides of the equation.


Remember that when taking the square root on the right-hand side the answer can be positive or negative.

x - 3 = ± 2

Step 7: Solve for x.

x = 3 ± 2


x = 5 or x = 1


Example 1: x2 + 2x - 5 = 0

Step 1: Write the equation in the general form


ax2 + bx + c = 0.

This equation is already in the proper form.

x2 + 2x - 5 = 0

Step 2: Move c, the constant term, to the right-hand side of the equation.

c = -5


x2 + 2x = 5

Step 3: Divide b, the x -coefficient, by two and square the result.

b = 2


2 2 =1r


(1)2 = 1

Step 4: Add the result from Step 3 to both sides of the equation.

x2 + 2x + 1 = 5 + 1

Step 5: Rewrite the left-hand side as a perfect square and simplify the right-hand side.


When rewriting in perfect square format the value in the parentheses is the b, x-coefficient, divided by 2 as found in Step 3.

(x + 1)2 = 5 + 1


(x + 1)2 = 6

Now that the square has been completed, solve for x.

Step 6: Take the square root of both sides of the equation.


Remember that when taking the square root on the right-hand side the answer can be positive or negative.

x+1=± 6


Step 7: Solve for x.

x=-1± 6


Example 2:      x 2 = 1 2 x

Step 1: Write the equation in the general form

ax2 + bx + c = 0.

x 2 1 2 x=0

Step 2: Move c, the constant term, to the right-hand side of the equation.


There is no c in this equation.

x 2 1 2 x=0

Step 3: Divide b, the x-coefficient, by two and square the result.

b= 1 2


1 2 ÷2= 1 4 r


( 1 4 ) 2 = 1 16

Step 4: Add the result from Step 3 to both sides of the equation.

x 2 1 2 x+ 1 16 =0+ 1 16

Step 5: Rewrite the left-hand side as a perfect square and simplify the right-hand side.


When rewriting in perfect square format the value in the parentheses is the b, x-coefficient, divided by 2 as found in Step 3.

( x 1 4 ) 2 =0+ 1 16


( x 1 4 ) 2 = 1 16

Now that the square has been completed, solve for x.

Step 6: Take the square root of both sides of the equation.


Remember that when taking the square root on the right-hand side the answer can be positive or negative.

x 1 4 =± 1 16

Step 7: Solve for x.

x= 1 4 ± 1 4


x= 1 2 or x=0


PART II of Completing the Square addresses the case when a ≠ 1.





Related Links:
Math
algebra
Completing the Square when a ≠ 1
Factoring Quadratic Equations when a equals 1
Algebra Topics


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