Question 607617: How do you solve 5x squared plus 16x minus 84 equals 0 using the quadratic equation
How do you solve 4x squared minus 12x plus 9 equals 0 using the quadratic equation
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! There are other ways to solve a quadratic equation, but using the quadratic formula always works if you are careful in doing your calculations.
SOLVING QUADRATIC EQUATIONS USING THE QUADRATIC FORMULA:
A quadratic equation is an equation of the form
 where the coefficients b, and c are any number and a is not zero.
Its solutions are given by the quadratic formula:
To solve a quadratic equation by using the quadratic formula, you need to determine the values for a, b, and c in the equation you are solving. You then plug those values into the formula, and calculate very carefully.
FOR  , a=5, b=16, and c=-84,
so 
The answers are
 and 
FOR  , a=4, b=-12, and c=9,
so 
The one and only answer is
THE OTHER WAYS TO SOLVE THOSE EQUATIONS:
Factoring, with some work, we get  ,
so the equation turns into
 with solutions when  --> 
and when  --> 
The factoring is much easier for , because it is obvious that  . It's the square of a binomial, one of the special products you are taught about in polynomials.
The only solution for  is the value for x that makes
 --> -->  -->  .
FACTORING  :
If it can be factored, it will be
 so we know that:
 so d and f have different signs, and
two factors (one positive and one negative) whose product is  will be such that 
There are many pairs of factors to 420 (we can worry about the signs later):
420=(1)(420)=(2)(210)=(3)(140)=(4)(105)=(5)(84)=(6)(70)=(7)(60)=(10)(42)=(12)(35)=(14)(30)=(15)(28)=(20)(21),
but the only pair that adds up to 16 (we consider sign now) is 30 and -14 (30-14=16).
So, before collecting like term, the product could have been
 and that is easier to factor by grouping, as follows:
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