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Completing the Square

Completing the Square

Author: Sophia Tutorial
Description:

Determine an equivalent quadratic equation by completing the square.

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Tutorial
what's covered
  1. Solving a Quadratic in the form (x+a)2
  2. FOIL and Completing the Square
  3. Solving a Quadratic by Completing the Square

1. Solving a Quadratic in the Form (x + a)2

Some quadratic equations are difficult to solve, and some quadratic equations are easy to solve. Depending on how the equation is provided, certain methods can be very straightforward. Consider the equation 5 = (x + 3)2. We can find solutions to x by performing inverse operations, just like we solve other multi-step equations:

5 equals open parentheses x plus 3 close parentheses squared
Our equation
plus-or-minus square root of 5 equals x plus 3
Take square root of both sides
short dash 3 plus-or-minus square root of 5 equals x
Subtract 3 from both sides
x equals short dash 3 minus square root of 5 comma space space space space x equals short dash 3 plus square root of 5
Our exact solutions
x equals short dash 5.24 comma space space space x equals short dash 0.76
Our approximate solutions

Unfortunately, not all quadratic equations come in this form. However, we can manipulate the equation to write it as such. This requires a process known as completing the square.


2. FOIL & Completing the Square

To understand the mechanics of completing the square, it is helpful to connect it to the FOIL process. Let's take the general expression (x + a)2 and FOIL it:

open parentheses x plus a close parentheses squared

open parentheses x plus a close parentheses open parentheses x plus a close parentheses
Two factors of open parentheses x plus a close parentheses
x squared plus a x plus a x plus a squared
FOIL
x squared plus 2 a x plus a squared
Combine like terms

big idea
The coefficient of the x-term is 2a, and the constant term is a2. If we can manipulate an expression to be in such a form, then we can write it equivalently as (x+a)2. This is the goal of completing the square.
term to know
Completing the Square
the process of converting a quadratic equation in the form a x squared plus b x plus c equals 0 into an expression involving a perfect square trinomial.


3. Solving a Quadratic by Completing the Square

The process of completing the square follows a specific set of steps in order to convert the equation into one similar to our very first example:

  • Move the constant term to the other side of the equation
  • Divide the entire equation by the x-square coefficient
  • Separately, divide the x-term coefficient by two, then square it
  • Add this quantity to both sides of the equation
  • Write one side of the equation from expanded form into factored form, expressed as a single factor squared.

Let's apply these steps to the equation: 0 equals 2 x squared minus 12 x minus 14

0 equals 2 x squared minus 12 x minus 14
Our quadratic formula
14 equals 2 x squared minus 12 x
Add 14 to both sides
7 equals x squared minus 6 x
Divide equation by 2
short dash 6 rightwards arrow fraction numerator short dash 6 over denominator 2 end fraction equals short dash 3 rightwards arrow open parentheses short dash 3 close parentheses squared equals 9

Divide short dash 6 by 2, then square it
7 plus 9 equals x squared minus 6 x plus 9
Add 9 to both sides
16 equals x squared minus 6 x plus 9
Simplify the left side
16 equals open parentheses x minus 3 close parentheses squared
Write right side as binomial squared

hint
The last step is a bit tricky. We were able to write the right side of the equation as a binomial squared, because of the relationship between the coefficients and the constant term. When short dash 3 is doubled, it is equivalent to the x-term coefficient in expanded form, short dash 6, AND when it is squared, it equals to constant term in the expanded form, 9.

  Now that we have converted an equation from standard form into a binomial squared, we can solve this equation following the same procedure as our very first example:

16 equals open parentheses x minus 3 close parentheses squared
Our quadratic formula
plus-or-minus square root of 16 equals x minus 3
Take square root of both sides
plus-or-minus 4 equals x minus 3
Simplify the left side
3 plus-or-minus 4 equals x
Add 3 to both sides
x equals 3 minus 4 space a n d space x equals 3 plus 4
Create two equations due to plus-or-minus
x equals short dash 1 space a n d space x equals 7
Our solutions

summary
When solving a quadratic in the form (x+a)2, we can find the solutions by performing inverse operations. FOIL and completing the square is a method of solving a quadratic equation. Solving a quadratic by competing the square involves rewriting a quadratic equation as a perfect square trinomial so that it can be written in factored form as (x+a)2, where a and y are real numbers.

Terms to Know
Completing the Square

the process of converting a quadratic equation in the form ax^2+bx+c =0 into an expression involving a perfect square trinomial