02Fcompletingsquare

= Solving Quadratic Equations by Completing the Square =

Recall that if you cannot find a way to factorise a quadratic, you can always resort to completing the square.


 * Example 1 **

math . \qquad \text{Solve : } \; x^2 + 8x + 3 = 0 math


 * Solution:**

math \\ . \qquad x^2 + 8x + 3 = 0 \qquad \qquad \{ \textit{halve and then square the middle term} \} \\.\\ . \qquad x^2 + 8x + 16 - 16 + 3 = 0 \quad \qquad \{ \textit{first 3 terms are now a perfect square} \} \\.\\ . \qquad \big( x + 4 \big)^2 - 13 = 0 \\. \\ . \qquad \big( x + 4 \big)^2 - \big( \sqrt{13} \big)^2 = 0 \qquad \qquad \{ \textit{this is now a difference of two squares} \} \\. \\ math math . \qquad \big( x + 4 + \sqrt{13} \big) \big(x + 4 - \sqrt{13} \big) = 0 \qquad \{ \textit{now use NFL} \} \\.\\ . \qquad x + 4 + \sqrt{13} = 0 \qquad \textbf{OR} \qquad x + 4 - \sqrt{13} = 0 \\.\\ . \qquad x = -4 - \sqrt{13} \qquad \textbf{ OR } \qquad x = -4 + \sqrt{13} math

Some people use an alternate approach to solving equations in this form:


 * Example 2 **

math . \qquad \text{Solve : } \; x^2 - 6x - 1 = 0 math


 * Solution:**

math \\ . \qquad x^2 - 6x - 1 = 0 \qquad \qquad \{ \textit{halve and then square the middle term} \} \\.\\ . \qquad x^2 - 6x + 9 - 9 - 1 = 0 \quad \qquad \{ \textit{first 3 terms are a perfect square} \} \\.\\ . \qquad \big( x - 3 \big)^2 - 10 = 0 \qquad \qquad \{ \textit{add 10 to both sides} \} \\.\\ math math . \qquad \big( x - 3 \big)^2 = 10 \qquad \qquad \{ \textit{square root both sides -- don't forget the plus/minus} \} \\.\\ . \qquad x - 3 = \pm \sqrt{10} \qquad \qquad \{ \textit{add 3 to both sides} \} \\.\\ . \qquad x = 3 \pm \sqrt{10} math

If using this approach, it is very important to remember to include the plus/minus symbol when you take the square root.

Not all Quadratic Equations have solutions

Many quadratic equations simply don't have __any__ solutions that are real numbers
 * remember that a solution is a number that you can substitute into the equation and get the left side equaling the right side.

If using the completing the square method, you end up with a __**plus**__ between the two terms: ... ... (x + 3) 2 ** + ** 2 = 0

Then write: ** No Real Solutions ** and stop.


 * Example: **

math . \qquad \text{Solve : } \; x^2 + 10x + 30 = 0 math


 * Solution:**

math . \qquad x^2 + 10x + 30 = 0 \\. \\ . \qquad x^2 + 10x + 25 - 25 + 30 = 0 \\. \\ . \qquad \big( x + 5 \big)^2 + 5 = 0 \qquad \qquad \{ \textit{ plus between the two terms} \} \\. \\ . \qquad \textbf{ No Real Solutions } math

Notice that if you did continue with this, you would get:

math . \qquad \big( x + 5 \big)^2 = -5 \\. \\ . \qquad x + 5 = \pm \sqrt{-5} math

And we can't take the square root of a negative number
 * unless we are working with __imaginary numbers__
 * you will learn about imaginary numbers in Year 12 Specialist Maths

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