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Completing the square is a method for solving quadratic equations by converting the equation from the form to the form . This method is particularly useful for quadratic equations that are difficult to factor using regular factorization.
Remember that a perfect square trinomial follows the pattern . We use this pattern to transform quadratic equations into a more solvable form.
Not all quadratic equations are easily factored. For example, the equation cannot be easily factored using rational numbers because there are no two numbers that multiply to give and add up to .
In such cases, the completing the square method becomes an effective choice for finding the roots of the equation.
Here are the steps to solve a quadratic equation using the completing the square method:
Ensure the coefficient of is
If the coefficient of is not , divide the entire equation by the value of .
Example: For the equation
Move the Constant Term to the Right Side
Move the constant term to the right side of the equation.
Example: From the equation
Add the Square of Half the Coefficient of x to Both Sides
Add to both sides of the equation. This value is the square of half the coefficient of .
Example: For the equation
Factor the Left Side into a Perfect Square
The left side now has the form , which can be factored as .
Simplify the Right Side
Perform calculations on the right side to get a simpler form.
Example: For
Take the Square Root of Both Sides
To eliminate the square, take the square root of both sides.
Example: From the equation
Solve for the Unknown Value
Isolate the variable to find the roots of the equation.
Example: From
Let's solve the equation:
Step : The coefficient , so we proceed to the next step.
Step : Move the constant to the right side.
Step : Add the square of half the coefficient of to both sides.
Step : Factor the left side into a perfect square.
Step : Simplify the right side.
Step : Take the square root of both sides.
Step : Solve for the value of .
Therefore, the roots of the equation are and .
Let's solve the equation:
Step : Divide all terms by the coefficient
Step : Move the constant to the right side
Step : Add the square of half the coefficient of to both sides
Step : Factor the left side into a perfect square
Step : Simplify the right side
Step : Take the square root of both sides
Step : Solve for the value of
Therefore, the roots of the equation are and .
For equations with coefficient of not equal to : Always divide the entire equation by the coefficient first. Example: becomes
Constant to be added: Always add the square of half the coefficient of to both sides. Example: For , add to both sides.
Final form: The equation will transform into the form . Example: becomes
If , then the equation has no real roots.
Concrete example:
Completing the square:
Since no real number has a square of , this equation has no real roots.
For equations of the form , we don't need to complete the square.
Concrete example:
Therefore, the roots of the equation are and .
Solve the following quadratic equations using the completing the square method:
Move the constant to the right side:
Add the square of half the coefficient of to both sides:
Factor the left side into a perfect square:
Simplify the right side:
Take the square root of both sides:
Solve for the value of :
Therefore, the roots of the equation are and .
Divide all terms by the coefficient :
Divide all terms by the coefficient :
Move the constant to the right side:
Divide all terms by the coefficient :
Half of the coefficient of is
The square of this value:
Add to both sides:
Example: From the equation
So the equation becomes:
For the positive sign:
For the negative sign:
Therefore, the roots of the equation are and .
Move the constant to the right side:
Add the square of half the coefficient of to both sides:
Factor the left side into a perfect square:
Simplify the right side:
Take the square root of both sides:
Solve for the value of :
Therefore, the roots of the equation are and .
Move the constant to the right side:
Add the square of half the coefficient of to both sides:
Factor the left side into a perfect square:
Simplify the right side:
Take the square root of both sides:
Solve for the value of :
Therefore, this equation has one (double) root, which is .
Add the square of half the coefficient of to both sides:
Factor the left side into a perfect square:
Take the square root of both sides:
Solve for the value of :
Therefore, the roots of the equation are and .
Move the constant to the right side:
Add the square of half the coefficient of to both sides:
Factor the left side into a perfect square:
Simplify the right side:
Take the square root of both sides:
Solve for the value of :
Therefore, the roots of the equation are and .
