SOLUTION: I need help on this one please 5x^2 = 8x+21

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: I need help on this one please 5x^2 = 8x+21      Log On


   

Question 870120: I need help on this one please
5x^2 = 8x+21
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
5x%5E2+=+8x%2B21 is a quadratic equation.
It would be customarily written in the more traditional form
5x%5E2+-+8x-21=0 .
There are several ways to solve it.
All quadratic equations can be solved two ways:
by "completing the square", and
by using the quadratic formula.
Some quadratic equations can also be solved by factoring.
Each method has its advantages and disadvantages,
some equations make one method look easier,
and people have their preferences for one or another method.

FACTORING sometimes is very easy.
It is not too easy for this equation, but it works, and it's what I would do.
The coefficients 5 and -21 (at the ends) multiply to give 5%2A%28-21%29=-105.
I will look for pairs of factors of -105 that add up to -8 , which is the middle coefficient.
105=1%2A105=3%2A35=5%2A21=7%2A15
Since 7%2A%2815%29=-105 and 7%2B%28-18%29=-8 , 7 and -15 will be the coefficients that I need.
I factor 5x%5E2-8x-21 "by parts" as follows:

Then I re-write the equation 5x%5E2-8x-21=0 as
%285x%2B7%29%28x-3%29=0 and realize that for the product to be zero, one factor must be zero,
so either x-3=0-->highlight%28x=3%29 ,
or 5x%2B7=0-->5x=-7-->highlight%28x=-7%2F5%29

COMPLETING THE SQUARE sometimes is very easy, it always works, and does not require remembering formulas.
It is not too easy in this case either, but here it goes:
5x%5E2-8x-21=0-->5x%5E2-8x=21-->%285x%5E2-8x%29%2F5=21%2F5-->x%5E2-%288%2F5%29x=21%2F5

so, from x%5E2-%288%2F5%29x=21%2F5 ,
adding %284%2F5%29%5E2=16%2F25 to both sides of the equal sign, I get the equivalent equation
x%5E2-%288%2F5%29x%2B%284%2F5%29%5E2=21%2F5%2B16%2F25 , which I can re-write as
%28x-4%2F5%29%5E2=105%2F25%2B16%2F25-->%28x-4%2F5%29%5E2=121%2F25-->%28x-4%2F5%29%5E2=%2811%2F5%29%5E2
So either x-4%2F5=11%2F5-->x=4%2F5%2B11%2F5-->x=15%2F5-->highlight%28x=3%29 ,
or x-4%2F5=-11%2F5-->x=4%2F5-11%2F5-->highlight%28x=-7%2F5%29

THE QUADRATIC FORMULA says that the solutions to an equation of the form
ax%5E2%2Bbx%2Bc=0 are given by
x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
That is easy enough if you
remember the formula,
have a calculator handy,
and have the knowledge and ability to use it without making mistakes.
In this case, we have 5x%5E2-8x-21=0 ,
so a=5, b=-8 and c=-21 .

The solutions are
x=%288%2B22%29%2F10=30%2F10=highlight%283%29
and x=%288-22%29%2F10=-14%2F10=highlight%28-7%2F5%29