Completing the Square Revision Notes for Grade 11 NSC Matric Mathematics

Completing the Square

What is completing the square?

Completing the square is a method for solving quadratic equations that cannot be easily factorised. This technique transforms a quadratic expression into a perfect square form, making it easier to solve.

The method works by creating a perfect square trinomial from the original quadratic equation. For example, we know that $(x - 1)^2 = x^2 - 2x + 1$ , so if we have an equation like $x^2 - 2x - 1 = 0$ , we can manipulate it to create this perfect square form.

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The name "completing the square" comes from the fact that we're literally completing a perfect square trinomial by adding the missing term needed to make it factorise as $(x + a)^2$ .

When do we use completing the square?

Completing the square is particularly useful when:

Factorisation is difficult or impossible
The quadratic equation doesn't factorise easily with integer factors
You need to find exact solutions that may be irrational
You want to write a quadratic in vertex form

Step-by-step method for completing the square

Follow these five key steps to complete the square:

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The 5-Step Method for Completing the Square

Step 1: Write the equation in standard form $ax^2 + bx + c = 0$

Step 2: Make the coefficient of the $x^2$ term equal to 1 by dividing the entire equation by $a$ (if $a \neq 1$ )

Step 3: Take half the coefficient of the $x$ term and square it, then add and subtract this value from the equation to keep it mathematically correct

Step 4: Write the trinomial as a perfect square

Step 5: Solve using one of two methods (explained below)

Two methods to solve after completing the square

Once you have completed the square and have an equation in the form $(x - h)^2 = k$ , you can solve using either method:

Method 1: Square root method

Take the square root of both sides and solve for $x$ . Remember to include both positive and negative solutions.

Method 2: Difference of two squares

Rewrite the equation as a difference of two squares and factorise: $(x - h)^2 - k = 0$ becomes $(x - h)^2 - (\sqrt{k})^2 = 0$ , which factorises to $[(x - h) + \sqrt{k}][(x - h) - \sqrt{k}] = 0$ .

Worked example 1: Rational solutions

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Worked Example: Solving with Rational Solutions

Question: Solve by completing the square: $x^2 - 10x - 11 = 0$

Solution:

Step 1: The equation is already in standard form $ax^2 + bx + c = 0$

Step 2: The coefficient of $x^2$ is already 1

Step 3: The coefficient of the $x$ term is $-10$ . Half of this is $-5$ , and $(-5)^2 = 25$ .

So we get: $x^2 - 10x + 25 - 25 - 11 = 0$

Step 4: Write as a perfect square: $(x - 5)^2 - 36 = 0$

Step 5: Using Method 1 (square roots):

$(x - 5)^2 = 36$
$x - 5 = \pm\sqrt{36}$
$x - 5 = \pm 6$
$x = 5 \pm 6$

Therefore: $x = 11$ or $x = -1$

Worked example 2: Irrational solutions

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Worked Example: Solving with Irrational Solutions

Question: Solve by completing the square: $2x^2 - 6x - 10 = 0$

Solution:

Step 1: The equation is in standard form

Step 2: Divide by 2 to make the coefficient of $x^2$ equal to 1:

$x^2 - 3x - 5 = 0$

Step 3: The coefficient of the $x$ term is $-3$ . Half of this is $-\frac{3}{2}$ , and $\left(-\frac{3}{2}\right)^2 = \frac{9}{4}$ .

So we get: $x^2 - 3x + \frac{9}{4} - \frac{9}{4} - 5 = 0$

Step 4: Write as a perfect square: $\left(x - \frac{3}{2}\right)^2 - \frac{29}{4} = 0$

Step 5: Using Method 1:

$\left(x - \frac{3}{2}\right)^2 = \frac{29}{4}$
$x - \frac{3}{2} = \pm\sqrt{\frac{29}{4}}$
$x - \frac{3}{2} = \pm\frac{\sqrt{29}}{2}$
$x = \frac{3}{2} \pm \frac{\sqrt{29}}{2}$

Therefore: $x = \frac{3 + \sqrt{29}}{2}$ or $x = \frac{3 - \sqrt{29}}{2}$

Note that these roots are irrational because 29 is not a perfect square.

Important exam tips and reminders

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Critical Points to Avoid Common Mistakes

Always remember the ± sign when taking square roots. Both positive and negative values are solutions.
When the coefficient of $x^2$ is not 1, you must divide the entire equation by this coefficient first.
The "half and square" rule is crucial: take half the coefficient of $x$ , then square it.
Some solutions will be rational (when you get a perfect square under the square root), others will be irrational.
You can check your answer by substituting back into the original equation.
Both the square root method and the difference of squares method will give the same solutions.

Remember!

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Key Points to Remember:

Completing the square is used when quadratic equations cannot be easily factorised
The key step is to "half and square" the coefficient of the $x$ term
Always include both positive and negative solutions when taking square roots
Make sure the coefficient of $x^2$ equals 1 before starting the process
Both solution methods (square roots and difference of squares) give the same answer

Completing the Square (Grade 11 NSC Matric Mathematics): Revision Notes