This week we were given back our quadratic equations test. From looking back, I learned that you can only take a part of an equation to simplify and do not have to use the entire equation or side of an equation that you are working on. What I mean by this is when taking a question like 5x²+25x-3=0. You can focus only on the 5x² and the positive 25x if there is a common multiple such as the number five in this example. Using this common multiple and recreating the equation, it would now look like 5(x²+5x)-3=0

This method helps to simplify the numbers down but to finish the equation you would need to make it a perfect square and add a zero pair. After dividing the 5x in the brackets by 2 and squaring it to get 25/4, you would put this into the equation, now making it look like 5(x²+5x+25⁄4-25⁄4)-3=0.

From here on, you would multiply the -25⁄4 to the 5 in order to get it out of the bracket to make – 125⁄4.

5(x²+5x+25⁄4)-125⁄4-3=0

Now, we are going to multiply all the numbers not over four by four on the denominator and the numerator to match the 25⁄4 and the -125⁄4.

20(4x²+20x+25)-125-12=0

Then, you can factor all the brackets to make 20(2x+5)² and add together the -125 and -12 to equal -137.

20(2x+5)²-137=0

We can then move the -137 to the other side of the brackets.

20(2x+5)²=137

After this, the equation needs to divide by twenty into both sides to get rid of the one in front of the brackets.

(2x+5)²=137⁄20

Then, square root both sides to get rid of the exponent and add an addition/subtraction sign to the 137⁄20.

2x+5=±√137⁄20

After moving over the five to the other side of the equal sign, you would make sure that there is nothing to simplify in the ±√137⁄20. The simplification you could accomplish is the twenty as the denominator. Which can be turned into 2√5.

2x=-5±√137/2√5

You must make the denominator have no roots by multiplying the top and bottom by √5.

2x=-5±√685/10

Then, to get rid of the 2 in front of x and leave it on its own, you divide all by 2.

x=-5⁄2±√685/20

From here on, you cannot simplify anymore and since there is an addition/subtraction sign, there are two answers to write down.

The other option to make this faster and easier in this specific example is to use the quadratic formula, but if you are specifically told to solve by a perfect square, or would like to go through the whole process. Then, this is how you can do it.