Question 584022: 12x^2-10x-42=0
I have to solve this quadratic equation two different ways and I need to show my work and explain what I am doing. I have no clue where to even start. I didn't even know there was two ways to solve this.....Any help please?
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website!  is equivalent to 
I would say there are 3 ways to solve it, and each person may have one preferred way.
You could:
factor the polynomial, or
use the quadratic formula, or
"complete the square."
FACTORING (not easy in this case):
Factoring is easier when the coefficient of x^2 is 1. It is easiest if the solutions and coefficients are all integers. It will not work if the solutions are not rational numbers.
We are looking for two factors whose product is . The products of factors like  or  will give you the terms  and  , but you need to get the term  right too.
If you are lucky, you may "see" the right factors immediately.
If not, there is a procedure.
Multiply the coefficient of  times the independent (constant) term, in this case, 
Look for pairs of factors for that product
-126=-(1)(126)=-(2)(63)=-(3)(42)=-(6)(21)=-(7)(18)=-(9)(14)
Find a pair of factors that adds to the coefficient of x. In this case, -14 and 9 work, because -14+9=-5).
Re-write the polynomial with two terms in x with the coefficients found.
Group the 4 terms in pairs with common factors.
Extract common factors, twice:
So is equivalent to and is equivalent to 
The solutions are what makes one or both factors zero:
 -->  --> 
 -->  --> 
QUADRATIC FORMULA (complicated, but always works)
An equation of the form  has the solutions  .
In this case a=6, b=-5, and c=-21, so
 -->  or
COMPLETING THE SQUARE
Transform the equation so that only the terms in  } and are on one side of the equal sign, and the term in is just .
-->  -->  --> 
Add to both sides the number that will make the left side a perfect square, and then write it as a square, and simplify:
 -->  -->  -->  --> 
If , either  or 
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