3 methods to finding roots

Okay, so today we were going over 3 different ways to calculate the roots of an equation. They were: factoring, completing the square and the quadratic formula.

Let's take the example:

5x²+7x+2=0

For the first method we factored the equation(you know how to do this).
You factor out like:
(5x+2)(x+1)=0
To factor an equation with a coefficient in front of the variable you expand it like so:

5x²+7x+2=0

5x²+5x+2x+2=0

5x(x+1)+2(x+1)=0

(5x+2)(x+1)=0

Now that you have that, we calculate the roots. For the first bracket, we isolate the 5x like so:
5x+2=0
5x= -2
Get x by itself to get:
x= -2/5

And this is your first root. The second is easier. Take (x+1) and figure out what number x can be to make it equal 0. That would make the second root, -1.
The two roots of 5x²+7x+2=0 are x= -2/5, -1.

The second method we did was Completing the Square(which we did earlier this week). It requires a little more time than the other methods but if done right, the results will be the same.
Let's take our example again:
5x²+7x+2=0

1. We factor out 5 to get x² on its own:
x²+7/5x+2/5=0

2. Now we focus on x²+7/5 and divide 7/5 by 2 and squaring it.
x²+7/5 --> Dividing by 2 in a fraction is like multiplying the denominator by 2. --> 7/5(1/2)
7/5 --> 7/10 then we square it --> (7/10)² = 49/100

3. We plug that into our equation as --> (x² + 7/5x + 49/100) +2/5 - 49/100=0
and factor it into --> (x+7/10)² +2/5 - 49/100=0

4. Now, let's transpose the last two terms so the equation is like this:
(x+7/10)² +2/5 - 49/100 --> (x+7/10)² = 49/100 - 2/5
Find the LCD between 49/100 and - 2/5 --> 49/100 - 40/100

5. Now you will have: (x+7/10)² = 9/100 so now you can square both sides:
√(x+7/10)² = + or-√9/100 (since it is a squared number, you can’t be sure if it is positive or negative)
x+7/10= + or - 3/10

6. Now you can calculate the roots--> transpose 7/10
--> x= -7/1o + 3/10 = -4/10= -2/5 <-- That is the first root.
x= -7/10 -3/10 = -10/10 = -1

The roots are x= -2/5, -1.

And lastly, Method No.3, the Quadratic formula which was also learned this week.
Once again, take 5x²+7x+2 and use the quadratic formula which is:
x= (-b +/-√b²-4ac) ÷ 2a

a= 5, b= 7, c= 2 Now plug that into the equation.

x= (-7 +/- (√7² - 4(5)(2)) ÷ 2(5) -->
x= -7 +/- (√49- 40) ÷ 10 -->
x= -7 +/- (√9) ÷ 10 -->
x= -7 +/- 3 ÷ 10 --> If it helps, you can look at it like: -7/10 +/- 3/10
Find the first root --> x= -7/10 + 3/10 = -4/10 = -2/5
The second root now --> x= -7/10 - 3/10 = -10/10 = -1

So after using all three methods, the roots will be x= -2/5, -1.

And thus is my blog for the day. If you have any problems or find my explanations a little hard to understand, feel free to make any comments. It was my first time so it might be a bit unorganized.

And now the next scribe will be… John…