Step-by-step explanation:
First, lets understand the question. When it says to find all the "real" solutions, that means you should not give any solutions containing an imaginary number. An imaginary number is basically where we accept that there exists an i (imaginary number) that is equal to [tex]\sqrt{-1\\}[/tex] (square root of negative 1). So we are not considering this concept in this question.
Let's start solving then. Can you think of an equation that may help you solve this problem? The biggest clue is that x is squared. If you said the quadratic formula, then you are perfectly correct!
Remember in order to use the quadratic formula, you must get your equation into the form:
[tex]ax^{2} +bx +c=0[/tex]
Let's do it! Right now we have:
[tex]5x^{2} - x +1=3[/tex]
Get rid of the 3 by subtracting it from both sides:
[tex]5x^{2} -x+1 -3 = 3 - 3\\5x^{2} -x-2=0[/tex]
Okay! it looks like our quadratic form now! By comparing it we can figure out what a, b, and c are in this case:
[tex]a = 5,{ }\\ b = -1, { }\\c = -2[/tex]
Let's get out our quadratic formula now:
[tex]\frac{ -b\frac{+}{} \sqrt{b^{2} -4ac} }{2a}[/tex]
All that's left to do is insert values and solve! (Be really careful when dealing with the negatives):
[tex]\frac{ -1\frac{+}{} \sqrt{(-1)^{2} -4(5)(-2)} }{2(5)}[/tex]
[tex]=\frac{ -1\frac{+}{} \sqrt{41} }{10}[/tex]
Removing the plus/minus we get two answers:
[tex]=\frac{ -1+ \sqrt{41} }{10}\\\\or\\\\=\frac{ -1-\sqrt{41} }{10}[/tex]
These are the most simplified forms! It turns out we didn't even have to worry about imaginary solutions because we didn't end up with a negative sign under the square root. Hope this helps, and good luck!
