Answer:
<u>x = 5 or 25</u>
Step-by-step explanation:
I think the method I am about to explain is slightly quicker and easier than the method in your question. This works for any 'complete the square' question.
We begin with x² - 30x = - 125.
First, we are going to factorise the left-hand side of the equation by <u>dividing the 'b' value </u><u>(-30)</u><u> by 2</u> (you'll see why this works in a minute):
(x - 15)²
We want these brackets to multiply out to give x² - 30x, so that they equal the left-hand side of the equation. Unfortunately, if we multiply them out, we get:
(x - 15)(x - 15) =
<u>x² - 30x + 225</u>
There is an <u>unwanted term</u> (the + 225, from 15²)! We only want x² - 30x, so we have to remove this term by <u>subtracting it from the left side of the equation</u>. To do this, let's set up the original equation again:
(x² - 30x + 225) <u>- 225</u> = - 125
<em>Note: </em><em>The reason why we don't have to subtract it from both sides is because the original equation is </em><em>x² - 30x = - 125</em><em>, and so we must make sure the left hand side is still equal to </em><em>x² - 30x</em><em>.</em>
So now we know that (x - 15)² multiplies out to give x² - 30x +225, we can write this as (x - 15)² in our equation:
(x² - 30x + 225) - 225 = - 125
is the same as:
(x - 15)² - 225 = - 125
Now add 225 to <u>both sides</u> of the equation:
(x - 15)² = - 125 + 225 = 100
(x -15)² = 100
The next step is to square root both sides. Be careful here, and remember that √100 can either be 10 or -10, as (-10)² = 100. To indicate both results, write ±10 ("plus or minus 10").
√(x - 15)² = √100
x - 15 = ±10
Because, there are two possible values for the right-hand side of the equation, we need to separate our equation into two equations:
1. x - 15 = 10
and
2. x - 15 = -10
Now we solve these two simple linear equations for x:
1. x = 10 + 15 <- add 15 to both sides
<u>x = 25</u> This is our first solution.
2. x = -10 + 15 <- add 15 to both sides again
<u>x = 5</u> This is our other solution.
<u>So our two solutions are x = 5 and x = 25!</u>
I have attached the quick version of the working out for this question - that is what you would be expected to write down in a test.
